On Relationships Between Two Linear Subspaces and Two Orthogonal

Spec. Matrices 2019; 7:142–212

Survey Article Open Access

Yongge Tian* On relationships between two linear subspaces and two orthogonal projectors https://doi.org/110.1515/spma-2019-0013 Received June 16, 2019; accepted September 22, 2019 Abstract: Sum and intersection of linear subspaces in a vector space over a eld are fundamental operations in linear algebra. The purpose of this survey paper is to give a comprehensive approach to the sums and intersections of two linear subspaces and their orthogonal complements in the nite-dimensional complex vector space. We shall establish a variety of closed-form formulas for representing the direct sum decompositions m of the m-dimensional complex column vector space C with respect to a pair of given linear subspaces M and N and their operations, and use them to derive a huge amount of decomposition identities for matrix expressions composed by a pair of orthogonal projectors onto the linear subspaces. As applications, we give matrix representation for the orthogonal projectors onto the intersections of a pair of linear subspaces using various matrix decomposition identities and Moore–Penrose inverses; necessary and sucient conditions for two linear subspaces to be in generic position; characterization of the commutativity of a pair of orthogonal projectors; necessary and sucient conditions for equalities and inequalities for a pair of subspaces to hold; equalities and inequalities for norms of a pair of orthogonal projectors and their operations; as well as a collection of characterizations of EP-matrix.

Keywords: Linear subspace, orthogonal complement, sum, intersection, orthogonal projector, matrix decomposition, Moore–Penrose inverse, generic position, commutativity, norm, EP matrix

MSC: 15A03, 15A09, 15A24, 15A27, 15B57, 47A15, 47H05

1 Introduction

Throughout this paper, we adopt the following notation: m×n m×n m R , C , and CH stand for the collections of all m × n real, complex matrices, and all m × m complex Hermitian matrices, respectively; m m×1 m m×1 C = C (R = R ) denotes the n-dimensional complex (real) vector space; r(A), R(A), and N (A) stand for the rank, the range (column space), and the kernel (null space) of a m×n matrix A ∈ C , respectively; Im denotes the identity matrix of order m; [ A, B ] stands for the columnwise partitioned matrix; A 0, A < 0, A ≺ 0, and A 4 0 mean that A is positive denite, positive semi-denite, negative denite, and negative semi-denite, respectively; m two A, B ∈ CH are said to satisfy the inequalities A B, A < B, A ≺ B, and A 4 B in the Löwner partial ordering if A − B is positive denite, positive semi-denite, negative denite, and negative semi-denite, respectively; m both i+(A) and i−(A), called the partial inertia of A ∈ CH , are dened to be the numbers of the positive and negative eigenvalues of A counted with multiplicities, respectively, where r(A) = i+(A) + i−(A);

*Corresponding Author: Yongge Tian: Shanghai Business School, Shanghai, China & Central University of Finance and Eco- nomics, Beijing, China, E-mail: [email protected]

m † the parallel sum of two positive semi-denite matrices A, B ∈ CH is dened to be A : B = A( A + B ) B; m×m 2 # the group inverse of A ∈ C with r(A ) = r(A), denoted by A , is dened to be the unique solution X satisfying the three matrix equations AXA = A, XAX = X and AX = XA;

the orthogonal projector onto M is a Hermitian idempotent matrix X, denoted by PM, that satises m range(X) = M; the collection of all orthogonal projectors of order m is denoted by COP; m×m ∗ ∗ m×m a complex matrix U ∈ C is unitary if U U = UU = Im; a Q ∈ R is call an orthogonal matrix if T T Q Q = QQ = Im. We next introduce denitions of generalized inverses of a matrix. The Moore–Penrose inverse of a given matrix m×n † n×m A ∈ C , denoted by A , is dened to be the unique matrix X ∈ C satisfying the four Penrose equations (i) AXA = A, (ii) XAX = X, (iii) (AX)∗ = AX, (iv) (XA)∗ = XA. (1.1)

In addition, a matrix X is called an {i, ... , j}-inverse of A, denoted by A(i,...,j), if it satises the ith,... , jth equations. The collection of all {i, ... , j}-inverses of A is denoted by {A(i,...,j)}. There are all 15 types of {i, ... , j}-inverse under (1.1), but the eight commonly-used generalized inverses of A induced from the rst † (1,3,4) (1,2,4) (1,2,3) (1,4) (1,3) (1,2) (1) † equation are given by A , A , A , A , A , A , A , and A . We also denote by PA = AA , † † EA = Im − AA , and FA = In − A A the three orthogonal projectors induced by A. Note from the denitions of generalized inverses of a matrix that they are in fact dened to be (common) solutions of some matrix equations. Thus analytical expressions of generalized inverses of matrix can be written as certain matrix-valued functions with one or more variable matrices. In fact, analytical formulas of generalized inverses of matrices and their functions are important issues and tools in matrix analysis, see e.g., [32, 48, 120]. Linear subspaces and orthogonal projectors onto linear subspaces are simplest and most generic concepts in linear algebra. These time-honored concepts are the parts of cornerstones in current mathematics, and are regarded as the very starting point of many advanced research topics in mathematics. Let M be a linear m subspace of the n-dimensional complex column vector space C . The orthogonal projector onto M is a Hermi- ∗ 2 tian idempotent matrix X = X = X , denoted by PM, that satises range(X) = M. Orthogonal projectors have many simple and nice properties from the viewpoint of matrices and operator theory, which can be characterized by many equations and facts; see, e.g., [75, 146]. On the other hand, orthogonal projectors onto linear subspaces are classic objects of study in linear vector spaces, and many properties of linear subspaces can be characterized by orthogonal projectors and their operations. Much attention has been paid to algebraic operations of two or more orthogonal projectors and their performance in various algebraic structures, and many results and facts on this topic can be found in the literature; see e.g., [1–4, 6, 9, 10, 12, 16, 17, 20–24, 27–30, 33– 35, 38, 41–43, 49–52, 63, 64, 69, 71, 73, 94–97, 99, 102, 110, 113, 115, 117, 119, 121, 131, 138, 145, 148, 149, 153]. As a summary, the aim of this paper is to provide a most detailed description on twists of two linear subspaces and two orthogonal projectors within the scope of linear algebra. The main content o this paper includes the following compilation of subjects: (I) establish simultaneous direct sum decomposition identities for two linear subspaces; (II) give many formulas for pure algebraic operations of two orthogonal projectors; (III) reformulate the CS decomposition for the pair of orthogonal projectors, and use it to derive many identities for the Moore–Penrose inverses of two orthogonal projectors and their algebraic operations; (IV) give four groups of matrix representation for the four orthogonal projectors onto the intersection subspaces, and present a group of identifying conditions for two linear subspaces to be in generic position; (V) give a collection of equivalent statements for two orthogonal projectors to be commutative; (VI) give a collection of equivalent statements for subspace equalities and inequalities to hold; (VII) give equalities and inequalities for representing the F-norms of two orthogonal projectors and their operations; (VIII) give collections of equivalent statements for a square matrix to be EP. A huge amount of formulas, equalities, inequalities, and facts are presented in 8 sections below covering the above topics, which will update our total understanding to linear subspaces and orthogonal projectors from all aspects. Most of the contents collected in this paper are so elementary and informative that we believe that they can denitely serve as beautiful and unreplaceable constructive materials for future textbooks in linear algebra and matrix theory. 144 Ë Yongge Tian

2 Fundamentals on a pair of linear subspaces

In this section, we rst present some fundamental facts on nite-dimensional vector space. Let K be a eld (such as the real or complex number eld), and V a vector space over K, and let M and N be two linear subspaces in V. Recall in linear algebra that the sum and the intersection of M and N are dened to be

M + N = { x + y | x ∈ M and y ∈ N }, (2.1) M ∩ N = { x | both x ∈ M and x ∈ N }, (2.2) respectively, both of which are linear subspaces in V as well. In particular,

W = M ⊕ N ⇐⇒ W = M + N and M ∩ N = {0}, (2.3) where M ⊕ N is called the direct sum of M and N. Recall that the standard inner product of any two complex m m (real) column vectors x, y in C (R ) is dened by hx, yi = x∗y (hx, yi = xT y). (2.4)

Given the inner product, the orthogonality of two vectors x and y, denoted by x ⊥ y, is dened by the require- ment

hx, yi = 0, (2.5)

m and two linear subspaces M and N of C are said to be orthogonal, denoted by M ⊥ N, (2.6) if and only if x ⊥ y for all x ∈ M and y ∈ N. If M ⊥ N, it must be the case that M and N are disjoint: m m M ∩ N = {0}. Given a subspace M of C , the orthogonal complement of M of C is dened to be ⊥ m M = {y ∈ C | hx, yi = 0 for all x ∈ M}. (2.7) Recall that the dimension of a nite-dimensional vector space (linear subspace), denoted by dim(·), is the number of independent vectors required to span the space (linear subspace). A best-known basic formula linking the dimensions of the sum and the intersection of two subspaces M1 and M2 in nite-dimensional vector space V is

dim(M1 + M2) + dim(M1 ∩ M2) = dim(M1) + dim(M2), (2.8) see e.g., [105, 126]. This formula shows that the dimensions the sum and the intersection of two linear subspaces are related, thus it can be used to describe relationship between two linear subspaces, for instance,

dim(M1 + M2) = dim(M1) + dim(M2) ⇐⇒ M1 ∩ M2 = {0}. (2.9)

One of the examples of the linear spaces is the range (column space) of a given matrix. Now let M1 = R(A1) m×n1 and M2 = R(A2) be two linear subspaces spanned by the column vectors of two matrices A1 ∈ C and m×n2 A2 ∈ C , respectively. Then the dimension formula in (2.8) can alternatively be expressed as

r[ A1, A2 ] + dim[R(A1) ∩ R(A2)] = r(A1) + r(A2). (2.10)

These basic results and facts can be found in almost all textbooks of linear algebra, and play a central role in the theory of nite-dimensional vector space. An extension of (2.8) with symmetric pattern to a family m of linear subspaces M1, M2, ... , Mk in C has recently been established by the present author in [139] as follows

(k − 1) dim(M1 + ··· + Mk) + dim(Mb 1 ∩ ··· ∩ Mb k) = dim(Mb 1) + ··· + dim(Mb k), (2.11) where Mb i = M1 + ··· + Mi−1 + Mi+1 + ··· + Mk , i = 1, 2, ... , k. In addition, we need to use in the sequel the following known results and facts; see e.g., [81, 93, 105, 126]. Linear subspaces and orthogonal projectors Ë 145

m Lemma 2.1. Let M and N be two linear subspaces in C . Then the following results hold. (a) M and its orthogonal complement obey the following rules

⊥ ⊥ ⊥ m ⊥ (M ) = M, M ⊕ M = C , dim M + dim M = m. (2.12)

In particular,

m ⊥ ⊥ m (C ) = {0} and {0} = C . (2.13)

(b) The following results hold

( M + N )⊥ = M⊥ ∩ N⊥, (M ∩ N)⊥ = M⊥ + N⊥, (2.14) M ⊆ N ⇔ M⊥ ⊇ N⊥. (2.15)

m Lemma 2.2. Let M1, M2, and M3 be three linear subspaces of C . Then the following results hold. (a) M1, M2, and M3 satisfy the distributive inequalities

M1 ∩ ( M2 + M3 ) ⊇ (M1 ∩ M2) + (M1 ∩ M3), (2.16)

M1 + ( M2 ∩ M3 ) ⊆ (M1 + M2) ∩ (M1 + M3), (2.17)

or equivalently, there exist two subspaces M0 and M00, called residual subspaces, such that

0 M1 ∩ ( M2 + M3 ) = [ (M1 ∩ M2) + (M1 ∩ M3)] ⊕ M , (2.18) 00 [M1 + ( M2 ∩ M3 )] ⊕ M = (M1 + M2) ∩ (M1 + M3). (2.19)

(b) The following modular identities hold

M1 ⊇ M2 ⇔ M1 ∩ ( M2 + M3 ) = M2 + (M1 ∩ M3), (2.20)

both M1 ⊇ M2 and M1 ⊇ M3 ⇔ M1 ∩ ( M2 + M3 ) = M2 + M3. (2.21)

⊥ ⊥ ⊥ ⊥ M1 ⊇ M2 ⇔ M1 = (M1 ∩ M2 ) ⊕ M2 ⇔ M1 = (M1 + M2) ∩ M2 . (2.22)

(d) The following symmetricity of commutativity holds

⊥ ⊥ M1 = (M1 ∩ M2) ⊕ (M1 ∩ M2 ) ⇔ M2 = (M1 ∩ M2) ⊕ (M1 ∩ M2). (2.23)

Assume that two linear subspaces in a vector space are given. One of the basic research problems in a vector space theory is concerned with the decompositions of the vector space with respect to the pair of subspaces. This kind of problem was rst proposed and approached in [80]; see also [42, 43, 66, 71, 93]. Now assume that m M and N are two linear subspaces of C . Then there are four intersections

M ∩ N, M ∩ N⊥, M⊥ ∩ N, M⊥ ∩ N⊥ (2.24) induced from M, N, M⊥, and N⊥. By (2.8), a set of dimension formulas associated with the above four intersections are given below.

m Lemma 2.3. Let M and N be two linear subspaces in C . Then

dim(M + N) = dim(M) + dim(N) − dim(M ∩ N), dim(M + N⊥) = dim(M) + dim(N⊥) − dim(M ∩ N⊥), dim(M⊥ + N) = dim(M⊥) + dim(N) − dim(M⊥ ∩ N), dim(M⊥ + N⊥) = dim(M⊥) + dim(N⊥) − dim(M⊥ ∩ N⊥). 146 Ë Yongge Tian

m Furthermore, some well-known decompositions of M, N, and C with respect to (2.24) are formulated follows.

m Lemma 2.4. Let M and N be two linear subspaces in C . Then m ⊥ ⊥ M = M ∩ C = M ∩ (N ⊕ N ) = (M ∩ N) ⊕ (M ∩ N ) ⊕ rest, (2.25) m ⊥ ⊥ N = C ∩ N = (M ⊕ M ) ∩ N = (M ∩ N) ⊕ (M ∩ N) ⊕ rest, (2.26) M + N = (M ∩ N) ⊕ (M ∩ N⊥) ⊕ (M⊥ ∩ N) ⊕ rest, (2.27) m ⊥ ⊥ ⊥ ⊥ C = (M ∩ N) ⊕ (M ∩ N ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊕ rest. (2.28) m The whole space C can be decomposed into the following four forms m ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ C = ( M + N ) ⊕ ( M + N ) = (M ∩ N ) ⊕ (M ∩ N ) = (M ∩ N ) ⊕ ( M + N ), (2.29) m ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ C = (M + N) ⊕ (M + N) = (M ∩ N ) ⊕ (M ∩ N ) = (M ∩ N ) ⊕ (M + N), (2.30) m ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ C = (M + N ) ⊕ (M + N ) = (M ∩ N) ⊕ (M ∩ N) = (M ∩ N) ⊕ (M + N ), (2.31) m ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ C = (M + N ) ⊕ (M + N ) = (M ∩ N) ⊕ (M ∩ N) = (M ∩ N) ⊕ (M + N ). (2.32) The following direct sum decomposition hold

m ⊥ ⊥ M = M ∩ C = M ∩ [(M ∩ N) ⊕ (M ∩ N) ] = (M ∩ N) ⊕ [M ∩ (M ∩ N) ], (2.33) m ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ M = M ∩ C = M ∩ [(M ∩ N ) ⊕ (M ∩ N ) ] = (M ∩ N ) ⊕ [M ∩ (M ∩ N ) ], (2.34) m ⊥ ⊥ N = N ∩ C = N ∩ [(M ∩ N) ⊕ (M ∩ N) ] = (M ∩ N) ⊕ [N ∩ (M ∩ N) ], (2.35) m ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ N = N ∩ C = N ∩ [(M ∩ N) ⊕ (M ∩ N) ] = (M ∩ N) ⊕ [N ∩ (M ∩ N) ], (2.36) ⊥ ⊥ m ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ M = M ∩ C = M ∩ [(M ∩ N) ⊕ (M ∩ N) ] = (M ∩ N) ⊕ [M ∩ (M ∩ N) ], (2.37) ⊥ ⊥ m ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ M = M ∩ C = M ∩ [(M ∩ N ) ⊕ (M ∩ N ) ] = (M ∩ N ) ⊕ [M ∩ (M ∩ N ) ], (2.38) ⊥ ⊥ m ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ N = N ∩ C = N ∩ [(M ∩ N ) ⊕ (M ∩ N) ] = (M ∩ N ) ⊕ [N ∩ (M ∩ N ) ], (2.39) ⊥ ⊥ m ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ N = N ∩ C = N ∩ [(M ∩ N ) ⊕ (M ∩ N ) ] = (M ∩ N ) ⊕ [N ∩ (M ∩ N ) ]. (2.40)

The decompositions in (2.25)–(2.28) and the analytical expressions of the rest components were approached, e.g., in [42, 43, 66, 71, 80, 93]. Eqs. (2.29)–(2.32) follow from (2.12)–(2.14). Eqs. (2.33)–(2.40) follow from (2.12) and (2.20). In the remaining of this section, we give a variety of novel formulas and facts on (2.25)–(2.28) and their variations. For convenience of representation, we dene a binary operation of subspaces M1 and M2 as follows

4 ⊥ ⊥ ⊥ M1 ◦ M2 = M1 ∩ (M1 ∩ M2 ) = M1 ∩ (M1 + M2). In term of this notation, we have the following eight expressions:

M ◦ N = M ∩ (M ∩ N⊥)⊥ = M ∩ (M⊥ + N), (2.41) N ◦ M = N ∩ (M⊥ ∩ N)⊥ = N ∩ (M + N⊥), (2.42) M ◦ N⊥ = M ∩ (M ∩ N)⊥ = M ∩ (M⊥ + N⊥), (2.43) N⊥ ◦ M = N⊥ ∩ (M⊥ ∩ N⊥)⊥ = N⊥ ∩ (M + N), (2.44) M⊥ ◦ N = M⊥ ∩ (M⊥ ∩ N⊥)⊥ = M⊥ ∩ (M + N), (2.45) N ◦ M⊥ = N ∩ (M ∩ N)⊥ = N ∩ (M⊥ + N⊥), (2.46) M⊥ ◦ N⊥ = M⊥ ∩ (M⊥ ∩ N)⊥ = M⊥ ∩ (M + N⊥), (2.47) N⊥ ◦ M⊥ = N⊥ ∩ (M ∩ N⊥)⊥ = N⊥ ∩ (M⊥ + N). (2.48)

Correspondingly,

(M ◦ N)⊥ = M⊥ ⊕ (M ∩ N⊥), (N ◦ M)⊥ = N⊥ ⊕ (M⊥ ∩ N), Linear subspaces and orthogonal projectors Ë 147

(M ◦ N⊥)⊥ = M⊥ ∩ (M ∩ N), (N⊥ ◦ M)⊥ = N ⊕ (M⊥ ∩ N⊥), (M⊥ ◦ N)⊥ = M ⊕ (M⊥ ∩ N⊥), (N ◦ M⊥)⊥ = N⊥ ⊕ (M ∩ N), (M⊥ ◦ N⊥)⊥ = M ⊕ (M⊥ ∩ N), (N⊥ ◦ M⊥)⊥ = N ⊕ (M ∩ N⊥).

Substituting (2.41)–(2.48) into (2.33)–(2.40), we obtain a group of simultaneous direct sum decomposition identities as follows.

m Theorem 2.5. Let M and N be two linear subspaces of C . Then the following results hold. (a) M, N, M⊥, and N⊥ can simultaneously be decomposed as

M = (M ∩ N) ⊕ (M ◦ N⊥) = (M ∩ N⊥) ⊕ (M ◦ N), N = (M ∩ N) ⊕ (N ◦ M⊥) = (M⊥ ∩ N) ⊕ (N ◦ M), M⊥ = (M⊥ ∩ N) ⊕ (M⊥ ◦ N⊥) = (M⊥ ∩ N⊥) ⊕ (M⊥ ◦ N), N⊥ = (M ∩ N⊥) ⊕ (N⊥ ◦ M⊥) = (M⊥ ∩ N⊥) ⊕ (N⊥ ◦ M).

(b) The following disjoint equalities hold

(M ◦ N⊥) ∩ (N ◦ M⊥) = {0}, (M ◦ N) ∩ (N⊥ ◦ M⊥) = {0}, (M⊥ ◦ N⊥) ∩ (N ◦ M) = {0}, (M⊥ ◦ N) ∩ (N⊥ ◦ M) = {0}.

M + N = (M ∩ N) ⊕ (M ◦ N⊥) ⊕ (N ◦ M⊥), M + N⊥ = (M ∩ N⊥) ⊕ (M ◦ N) ⊕ (N⊥ ◦ M⊥), M⊥ + N = (M⊥ ∩ N) ⊕ (M⊥ ◦ N⊥) ⊕ (N ◦ M), M⊥ + N⊥ = (M⊥ ∩ N⊥) ⊕ (M⊥ ◦ N) ⊕ (N⊥ ◦ M), m ⊥ ⊥ ⊥ ⊥ C = (M ∩ N ) ⊕ (M ∩ N ) ⊕ (M ◦ N) ⊕ (M ◦ N), m ⊥ ⊥ ⊥ ⊥ C = (M ∩ N) ⊕ (M ∩ N ) ⊕ (N ◦ M) ⊕ (N ◦ M), m ⊥ ⊥ ⊥ ⊥ C = (M ∩ N) ⊕ (M ∩ N ) ⊕ (M ◦ N ) ⊕ (N ◦ M ), m ⊥ ⊥ ⊥ ⊥ C = (M ∩ N ) ⊕ (M ∩ N) ⊕ (M ◦ N) ⊕ (N ◦ M ), m ⊥ ⊥ ⊥ ⊥ C = (M ∩ N) ⊕ (M ∩ N ) ⊕ (M ◦ N ) ⊕ (N ◦ M), m ⊥ ⊥ ⊥ ⊥ C = (M ∩ N) ⊕ (M ∩ N ) ⊕ (M ◦ N) ⊕ (N ◦ M). (d) [77] The following direct sum decomposition identities hold

m ⊥ ⊥ ⊥ ⊥ C = (M ◦ N) ⊕ (N ◦ M) = (M ◦ N) ⊕ (N ◦ M ) = (M ◦ N⊥) ⊕ (N⊥ ◦ M)⊥ = (M⊥ ◦ N⊥) ⊕ (N⊥ ◦ M⊥)⊥.

(e) The following dimension formulas hold

dim(M) = dim(M ∩ N) + dim(M ◦ N⊥) = dim(M ∩ N⊥) + dim(M ◦ N), dim(N) = dim(M ∩ N) + dim(N ◦ M⊥) = dim(M⊥ ∩ N) + dim(N ◦ M), dim(M⊥) = dim(M⊥ ∩ N) + dim(M⊥ ◦ N⊥) = dim(M⊥ ∩ N⊥) + dim(M⊥ ◦ N), dim(N⊥) = dim(M ∩ N⊥) + dim(N⊥ ◦ M⊥) = dim(M⊥ ∩ N⊥) + dim(N⊥ ◦ M), dim(M + N) = dim(M ∩ N) + dim(M ◦ N⊥) + dim(N ◦ M⊥), dim(M + N⊥) = dim(M ∩ N⊥) + dim(M ◦ N) + dim(N⊥ ◦ M⊥), dim(M⊥ + N) = dim(M⊥ ∩ N) + dim(M⊥ ◦ N⊥) + dim(N ◦ M), dim(M⊥ + N⊥) = dim(M⊥ ∩ N⊥) + dim(M⊥ ◦ N) + dim(N⊥ ◦ M). 148 Ë Yongge Tian

For convenience of representing the residuals in (2.25)–(2.28), we dene a binary operation for two subspaces as follows

4 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ M1 • M2 = M1 ∩ (M1 ∩ M2) ∩ (M1 ∩ M2 ) = M1 ∩ (M1 + M2 ) ∩ (M1 + M2).

In term of this notation, we obtain

M • N = M • N⊥ = M ∩ (M ∩ N)⊥ ∩ (M ∩ N⊥)⊥, (2.49) N • M = N • M⊥ = N ∩ (M ∩ N)⊥ ∩ (M⊥ ∩ N)⊥, (2.50) M⊥ • N = M⊥ • N⊥ = M⊥ ∩ (M⊥ ∩ N)⊥ ∩ (M⊥ ∩ N⊥)⊥, (2.51) N⊥ • M = N⊥ • M⊥ = N⊥ ∩ (M ∩ N⊥)⊥ ∩ (M⊥ ∩ N⊥)⊥. (2.52)

Correspondingly,

(M • N)⊥ = (M • N⊥)⊥ = M⊥ + (M ∩ N) + (M ∩ N⊥), (2.53) (N • M)⊥ = (N • M⊥)⊥ = N⊥ + (M ∩ N) + (M⊥ ∩ N), (2.54) (M⊥ • N)⊥ = (M⊥ • N⊥)⊥ = M + (M⊥ ∩ N) + (M⊥ ∩ N⊥), (2.55) (N⊥ • M)⊥ = (N⊥ • M⊥)⊥ = N + (M ∩ N⊥) + (M⊥ ∩ N⊥). (2.56)

By using the above expressions, we obtain below another group of simultaneous direct sum decomposition identities and their algebraic properties.

m Theorem 2.6. Let M and N be two linear subspaces of C . Then the following results hold. (a) M, N, M⊥, and N⊥ can be decomposed as direct sums of three terms

M = (M ∩ N) ⊕ (M ∩ N⊥) ⊕ (M • N), N = (M ∩ N) ⊕ (M⊥ ∩ N) ⊕ (N • M), M⊥ = (M⊥ ∩ N) ⊕ (M⊥ ∩ N⊥) ⊕ (M⊥ • N), N⊥ = (M ∩ N⊥) ⊕ (M⊥ ∩ N⊥) ⊕ (N⊥ • M).

Alternatively, M, N, M⊥, and N⊥ can be decomposed as intersections of three terms

M = (M + N) ∩ (M + N⊥) ∩ (M⊥ • N)⊥, N = (M + N) ∩ (M⊥ ∩ N) ∩ (N⊥ • M)⊥, M⊥ = (M⊥ + N⊥) ∩ (M⊥ + N) ∩ (M • N)⊥, N⊥ = (M⊥ + N⊥) ∩ (M ∩ N⊥) ∩ (N • M)⊥,

where

(M • N)⊥ = M⊥ + (M ∩ N) + (M ∩ N⊥), (N • M)⊥ = N⊥ + (M ∩ N) + (M⊥ ∩ N), (M⊥ • N)⊥ = M + (M⊥ ∩ N)⊥ + (M⊥ + N⊥), (N⊥ • M)⊥ = N + (M ∩ N⊥) + (M⊥ ∩ N⊥).

(b) The following disjoint facts hold

[(M ∩ N⊥) ⊕ (M • N)] ∩ [M⊥ ∩ N) ⊕ (N • M)] = {0}, [(M ∩ N) ⊕ (M • N)] ∩ [(M⊥ ∩ N⊥) ⊕ (N⊥ • M)] = {0}, [(M⊥ ∩ N⊥) ⊕ (M⊥ • N)] ∩ [(M ∩ N) ⊕ (N • M)] = {0}, [(M⊥ ∩ N) ⊕ (M⊥ • N)] ∩ [(M ∩ N⊥) ⊕ (N⊥ • M)] = {0}.

PM = PM∩N + PM∩N⊥ + PM•N, PN = PM∩N + PM⊥∩N + PN•M,

PM⊥ = PM⊥∩N + PM⊥∩N⊥ + PM⊥•N, PN⊥ = PM∩N⊥ + PM⊥∩N⊥ + PN⊥•M;

the following identities hold

PMPN = PM∩N + PM•NPN•M, PNPM = PM∩N + PN•MPM•N, Linear subspaces and orthogonal projectors Ë 149

PMPN⊥ = PM∩N⊥ + PM•NPN⊥•M, PN⊥ PM = PM∩N⊥ + PN⊥•MPM•N,

PM⊥ PN = PM⊥∩N + PM⊥•NPN•M, PNPM⊥ = PM⊥∩N + PN•MPM⊥•N,

PM⊥ PN⊥ = PM⊥∩N⊥ + PM⊥•NPN⊥•M, PN⊥ PM⊥ = PM⊥∩N⊥ + PN⊥•MPM⊥•N;

the following identities hold

PMPN − PNPM = PM•NPN•M − PN•MPM•N,

PMPN⊥ − PN⊥ PM = PM•NPN⊥•M − PN⊥•MPM•N,

PM⊥ PN − PNPM⊥ = PM⊥•NPN•M − PN•MPM⊥•N,

PM⊥ PN⊥ − PN⊥ PM⊥ = PM⊥•NPN⊥•M − PN⊥•MPM⊥•N,

PMPN − PNPM = PN⊥ PM − PMPN⊥ = PN⊥ PM − PMPN⊥ = PM⊥ PN⊥ − PN⊥ PM⊥ ;

and the following commutative equalities hold

(PM − PM•N)(PN − PN•M) = (PN − PN•M)(PM − PM•N),

(PM − PM•N)(PN⊥ − PN⊥•M) = (PN⊥ − PN⊥•M)(PM − PM•N),

(PM⊥ − PM⊥•N)(PN − PN•M) = (PN − PN•M)(PM⊥ − PM⊥•N),

(PM⊥ − PM⊥•N)(PN⊥ − PN⊥•M) = (PN⊥ − PN⊥•M)(PM⊥ − PM⊥•N).

(d) The following dimension formulas hold

dim(M) = dim(M ∩ N) + dim(M ∩ N⊥) + dim(M • N), dim(N) = dim(M ∩ N) + dim(M⊥ ∩ N) + dim(M • N), dim(M⊥) = dim(M⊥ ∩ N) + dim(M⊥ ∩ N⊥) + dim(M⊥ • N), dim(N⊥) = dim(M ∩ N⊥) + dim(M⊥ ∩ N⊥) + dim(N⊥ • M), dim(M) + dim(M⊥ ∩ N) = dim(N) + dim(M ∩ N⊥), dim(M) + dim(M⊥ ∩ N⊥) = dim(N⊥) + dim(M ∩ N), dim(M⊥) + dim(M ∩ N) = dim(N) + dim(M⊥ ∩ N⊥), dim(M⊥) + dim(M ∩ N⊥) = dim(N⊥) + dim(M⊥ ∩ N), dim(M • N) = dim(N • M) = dim(M⊥ • N) = dim(N⊥ • M).

(e) The following ve-term simultaneous direct sum decomposition identities hold

M + N = (M ∩ N) ⊕ (M ∩ N⊥) ⊕ (M⊥ ∩ N) ⊕ (M • N) ⊕ (N • M), M + N⊥ = (M ∩ N) ⊕ (M ∩ N⊥) ⊕ (M⊥ ∩ N⊥) ⊕ (M • N) ⊕ (N⊥ • M), M⊥ + N = (M ∩ N) ⊕ (M⊥ ∩ N) ⊕ (M⊥ ∩ N⊥) ⊕ (M⊥ • N) ⊕ (N • M), M⊥ + N⊥ = (M ∩ N⊥) ⊕ (M⊥ ∩ N) ⊕ (M⊥ ∩ N⊥) ⊕ (M⊥ • N) ⊕ (N⊥ • M).

(f) The following orthogonal projector decomposition identities hold

PM+N = PM∩N + PM∩N⊥ + PM⊥∩N + P(M•N)+(N•M),

PM+N⊥ = PM∩N + PM∩N⊥ + PM⊥∩N⊥ + P(M•N)+(N⊥•M),

PM⊥+N = PM∩N + PM⊥∩N + PM⊥∩N⊥ + P(M⊥•N)+(N•M),

PM⊥+N⊥ = PM∩N⊥ + PM⊥∩N + PM⊥∩N⊥ + P(M⊥•N)+(N⊥•M).

(g) The following dimension identities hold

dim(M + N) = dim(M ∩ N) + dim(M ∩ N⊥) + dim(M⊥ ∩ N) + dim(M • N) + dim(N • M), 150 Ë Yongge Tian

dim(M + N⊥) = dim(M ∩ N) + dim(M ∩ N⊥) + dim(M⊥ ∩ N⊥) + dim(M • N) + dim(N⊥ • M), dim(M⊥ + N) = dim(M ∩ N) + dim(M⊥ ∩ N) + dim(M⊥ ∩ N⊥) + dim(M⊥ • N) + dim(N • M), dim(M⊥ + N⊥) = dim(M ∩ N⊥) + dim(M⊥ ∩ N) + dim(M⊥ ∩ N⊥) + dim(M⊥ • N) + dim(N⊥ • M), dim(M + N) + dim(N⊥) = dim(M⊥ + N⊥) + dim(M), dim(M + N⊥) + dim(N) = dim(M⊥ + N) + dim(M), dim(M⊥ + N) + dim(N⊥) = dim(M + N⊥) + dim(M⊥), dim(M⊥ + N⊥) + dim(N) = dim(M + N) + dim(M⊥).

Proof. Note from (2.12) that

⊥ ⊥ ⊥ m [(M ∩ N) ⊕ (M ∩ N )] ⊕ [(M ∩ N) ⊕ (M ∩ N )] = C , (2.57) M ⊇ (M ∩ N) ⊕ (M ∩ N⊥). (2.58)

Hence, we obtain from (2.20) that

M = M ∩ {[(M ∩ N) ⊕ (M ∩ N⊥)] ⊕ [(M ∩ N) ⊕ (M ∩ N⊥)]⊥} = [ (M ∩ N) ⊕ (M ∩ N⊥)] ⊕ {M ∩ [(M ∩ N) ⊕ (M ∩ N⊥)]⊥} = [ (M ∩ N) ⊕ (M ∩ N⊥)] ⊕ [M ∩ (M ∩ N)⊥ ∩ (M ∩ N⊥)⊥] = (M ∩ N) ⊕ (M ∩ N⊥) ⊕ (M • N), establishing the rst equality in (a), where three terms on the right-hand side are obviously pairwise orthogonal. The other three equalities in (a) can be shown similarly. Also note that

(M • N) ∩ (N • M) = (M ∩ N) ∩ (M ∩ N)⊥ ∩ (M ∩ N⊥)⊥ ∩ (M⊥ ∩ N)⊥ = {0}, establishing the rst identity in (b), so that

(M • N) + (N • M) = (M • N) ⊕ (N • M).

Adding the rst and second equalities in (a) and applying the above equality lead to the rst equality in (e). The remaining equalities in (e) can be established by a similar approach. Taking orthogonal projectors of both sides of (a) and applying orthogonality among the three terms on the right-hand sides leads to the rst group of equalities in (c). The second and third groups of equalities in

(c) follow from rst group of equalities in (c), PM⊥ = Im − PM and PN⊥ = Im − PN. Taking dimensions of both sides of (a) leads to the rst four equalities in (d), as well as the remaining equalities in (d). Taking orthogonal projectors of both sides of four equalities in (e) leads to (f). Taking dimensions of both sides of four equalities in (e) leads to (g).

We next derive more facts about two subspaces and their operations. The following concept on the commutator of two subspaces is well known; see e.g., [93].

m Denition 2.7. Let M and N be two linear subspaces of C . The commutator of M and N is dened to be Comm(M, N) = (M ∩ N) ⊕ (M ∩ N⊥) ⊕ (M⊥ ∩ N) ⊕ (M⊥ ∩ N⊥).

Substituting Theorem 2.6(e) into (2.29)–(2.32) and using Denition 2.7, we obtain the following consequences.

m Theorem 2.8. Let M and N be two linear subspaces of C . Then, the following results hold. (a) The dimension of Comm(M, N) is

dim[Comm(M, N)] = dim(M ∩ N) + dim(M ∩ N⊥) + dim(M⊥ ∩ N) + dim(M⊥ ∩ N⊥). Linear subspaces and orthogonal projectors Ë 151

m (b) C can be written as the following four ve-term direct sums

m ⊥ C = Comm(M, N) ⊕ (M • N) ⊕ (N • M) = Comm(M, N) ⊕ (M • N) ⊕ (N • M) = Comm(M, N) ⊕ (M⊥ • N) ⊕ (N • M) = Comm(M, N) ⊕ (M⊥ • N) ⊕ (N⊥ • M).

It can be seen from Theorems 2.6(a) and 2.8(b) that all possible ordinary operations of two subspaces M and m N in C can be decomposed as the direct sums of the following eight canonical subspaces:

M ∩ N, M ∩ N⊥, M⊥ ∩ N, M⊥ ∩ N⊥, M • N, N • M, M⊥ • N, M⊥ • N.

For convenience, we call the four seminal expansion formulas in Theorem 2.6(a) the joint direct sum decom- m position identities (JDSDIs) of two subspaces in C . The closed-form expressions of the four JDSDIs and their subsequences have a great advantage in dealing with various problems related to two subspaces, and a huge amount of consequences, as common exercises in linear algebra, can be derived from these fundamental JDSDIs only using usual symbolic operations of subspaces.

m Theorem 2.9. Let M and N be two linear subspaces of C . (a) The following results hold

M ◦ N = (M ∩ N) ⊕ (M • N), M ◦ N⊥ = (M ∩ N⊥) ⊕ (M • N), M⊥ ◦ N = (M⊥ ∩ N) ⊕ (M⊥ • N), M⊥ ◦ N⊥ = (M⊥ ∩ N⊥) ⊕ (M⊥ • N), N ◦ M = (N ∩ M) ⊕ (N • M), N ◦ M⊥ = (N ∩ M⊥) ⊕ (N • M), N⊥ ◦ M = (N⊥ ∩ M) ⊕ (N⊥ • M), N⊥ ◦ M⊥ = (N⊥ ∩ M⊥) ⊕ (N⊥ • M).

(b) The following results hold

(M ◦ N) ∩ (N ◦ M) = M ∩ N, (M ◦ N⊥) ∩ (N⊥ ◦ M) = M ∩ N⊥, (M⊥ ◦ N) ∩ (N ◦ M⊥) = M⊥ ∩ N, (M⊥ ◦ N⊥) ∩ (N⊥ ◦ M⊥) = M⊥ ∩ N⊥, (M ◦ N) ∩ (M ◦ N⊥) = M • N, (N ◦ M) ∩ (N ◦ M⊥) = N • M, (M⊥ ◦ N) ∩ (M⊥ ◦ N⊥) = M⊥ • N, (N⊥ ◦ M) ∩ (N⊥ ◦ M⊥) = N⊥ • M.

Proof. The eight equalities in (a) follow from comparing Theorem 2.5(a) with Theorem 2.6(a). It is easy to see

(M ◦ N) ∩ (N ◦ M) = (M ∩ N) ∩ (M ∩ N⊥)⊥ ∩ (M⊥ ∩ N)⊥ ⊆ M ∩ N. (2.59)

Also note from (2.16) that

(M ∩ N) ∩ (M ∩ N⊥)⊥ = (M ∩ N) ∩ (M⊥ + N) ⊇ (M ∩ N ∩ M⊥) + (M ∩ N ∩ N) = M ∩ N, (M ∩ N) ∩ (M⊥ ∩ N)⊥ = (M ∩ N) ∩ (M + N⊥) ⊇ (M ∩ N ∩ M) + (M ∩ N ∩ N⊥) = M ∩ N.

So that

(M ◦ N) ∩ (N ◦ M) = (M ∩ N) ∩ (M ∩ N⊥)⊥ ∩ (M⊥ ∩ N)⊥ ⊇ M ∩ N. (2.60)

Combining (2.59) and (2.60) gives the rst equality in (b). The second to fourth equalities in (b) can be shown similarly. The remaining four equalities follow directly from (2.41)–(2.48) and (2.49)–(2.52).

From Theorems 2.5 and 2.6, we further obtain the following result.

m Theorem 2.10. Let M and N be two linear subspaces of C . Then, the following results hold. (a) M, N, M⊥, and N⊥ can simultaneously be decomposed as direct sums of three terms

M = ((M ◦ N) ∩ (N ◦ M)) ⊕ ((M ◦ N⊥) ∩ (N⊥ ◦ M)) ⊕ ((M ◦ N) ∩ (M ◦ N⊥)), 152 Ë Yongge Tian

N = ((M ◦ N) ∩ (N ◦ M)) ⊕ ((M⊥ ◦ N) ∩ (N ◦ M⊥)) ⊕ ((N ◦ M) ∩ (N ◦ M⊥)), M⊥ = ((M⊥ ◦ N) ∩ (N ◦ M⊥)) ⊕ ((M⊥ ◦ N⊥) ∩ (N⊥ ◦ M⊥)) ⊕ ((M⊥ ◦ N) ∩ (M⊥ ◦ N⊥)), N⊥ = ((M ◦ N⊥) ∩ (N⊥ ◦ M)) ⊕ ((M⊥ ◦ N⊥) ∩ (N⊥ ◦ M⊥)) ⊕ ((N⊥ ◦ M) ∩ (N⊥ ◦ M⊥)).

Alternatively, M, N, M⊥, and N⊥ can be decomposed as intersections of three terms

M = ((M⊥ ◦ N)⊥ + (N ◦ M⊥)⊥) ∩ ((M⊥ ◦ N⊥)⊥ + (N⊥ ◦ M⊥)⊥)) ∩ ((M⊥ ◦ N)⊥ + (M⊥ ◦ N⊥)⊥), N = ((M ◦ N⊥)⊥ + (N⊥ ◦ M)⊥) ∩ ((M⊥ ◦ N⊥)⊥ + (N⊥ ◦ M⊥)⊥) ∩ ((N⊥ ◦ M)⊥ + (N⊥ ◦ M⊥)⊥), M⊥ = ((M ◦ N)⊥ + (N ◦ M)⊥) ∩ ((M ◦ N⊥)⊥ + (N⊥ ◦ M)⊥) ∩ ((M ◦ N)⊥ + (M ◦ N⊥)⊥), N⊥ = ((M ◦ N)⊥ + (N ◦ M)⊥) ∩ ((M⊥ ◦ N)⊥ + (N ◦ M⊥)⊥) ∩ ((N ◦ M)⊥ + (N ◦ M⊥)⊥).

(b) The sums M + N, M + N⊥, M⊥ + N, and M⊥ + N⊥ can simultaneously be written as ve-term direct sums

M + N = ((M ◦ N) ∩ (N ◦ M)) ⊕ ((M ◦ N⊥) ∩ (N⊥ ◦ M)) ⊕ ((M⊥ ◦ N) ∩ (N ◦ M⊥)) ⊕ ((M ◦ N) ∩ (M ◦ N⊥)) ⊕ ((N ◦ M) ∩ (N ◦ M⊥)),

M + N⊥ = ((M ◦ N) ∩ (N ◦ M)) ⊕ ((M ◦ N⊥) ∩ (N⊥ ◦ M)) ⊕ ((M⊥ ◦ N⊥) ∩ (N⊥ ◦ M⊥)) ⊕ ((M ◦ N) ∩ (M ◦ N⊥)) ⊕ ((N⊥ ◦ M) ∩ (N⊥ ◦ M⊥)), M⊥ + N = ((M ◦ N) ∩ (N ◦ M)) ⊕ ((M⊥ ◦ N) ∩ (N ◦ M⊥)) ⊕ ((M⊥ ◦ N⊥) ∩ (N⊥ ◦ M⊥)) ⊕ ((M⊥ ◦ N) ∩ (M⊥ ◦ N⊥)) ⊕ ((N ◦ M) ∩ (N ◦ M⊥)), M⊥ + N⊥ = ((M ◦ N⊥) ∩ (N⊥ ◦ M)) ⊕ ((M⊥ ◦ N) ∩ (N ◦ M⊥)) ⊕ ((M⊥ ◦ N⊥) ∩ (N⊥ ◦ M⊥)) ⊕ ((M⊥ ◦ N) ∩ (M⊥ ◦ N⊥)) ⊕ ((N⊥ ◦ M) ∩ (N⊥ ◦ M⊥)).

m (c) C can be decomposed as

m ⊥ ⊥ C = ((M ◦ N) ∩ (N ◦ M)) ⊕ ((M ◦ N ) ∩ (N ◦ M)) ⊕ ((M⊥ ◦ N) ∩ (N ◦ M⊥)) ⊕ ((M⊥ ◦ N⊥) ∩ (N⊥ ◦ M⊥)) ⊕ ((M ◦ N) ∩ (M ◦ N⊥)) ⊕ ((N ◦ M) ∩ (N ◦ M⊥)), m ⊥ ⊥ C = ((M ◦ N) ∩ (N ◦ M)) ⊕ ((M ◦ N ) ∩ (N ◦ M)) ⊕ ((M⊥ ◦ N) ∩ (N ◦ M⊥)) ⊕ ((M⊥ ◦ N⊥) ∩ (N⊥ ◦ M⊥)) ⊕ ((M ◦ N) ∩ (M ◦ N⊥)) ⊕ ((N⊥ ◦ M) ∩ (N⊥ ◦ M⊥)), m ⊥ ⊥ C = ((M ◦ N) ∩ (N ◦ M)) ⊕ ((M ◦ N ) ∩ (N ◦ M)) ⊕ ((M⊥ ◦ N) ∩ (N ◦ M⊥)) ⊕ ((M⊥ ◦ N⊥) ∩ (N⊥ ◦ M⊥)) ⊕ ((M⊥ ◦ N) ∩ (M⊥ ◦ N⊥)) ⊕ ((N ◦ M) ∩ (N ◦ M⊥)), m ⊥ ⊥ C = ((M ◦ N) ∩ (N ◦ M)) ⊕ ((M ◦ N ) ∩ (N ◦ M)) ⊕ ((M⊥ ◦ N) ∩ (N ◦ M⊥)) ⊕ ((M⊥ ◦ N⊥) ∩ (N⊥ ◦ M⊥)) ⊕ ((M⊥ ◦ N) ∩ (M⊥ ◦ N⊥)) ⊕ ((N⊥ ◦ M) ∩ (N⊥ ◦M⊥)). Linear subspaces and orthogonal projectors Ë 153

Concerning relationship between two linear subspaces, the following two denitions are well known.

m Denition 2.11. A pair of linear subspaces M and N in C are said to be in generic position if and only if the following four equalities hold

M ∩ N = {0}, M ∩ N⊥ = {0}, M⊥ ∩ N = {0}, M⊥ ∩ N⊥ = {0}.

m Denition 2.12. A pair of linear subspaces M and N in C are said to be commutative if and only if

M = (M ∩ N) ⊕ (M ∩ N⊥) and/or N = (M ∩ N) ⊕ (M⊥ ∩ N).

The concept of generic position was introduced in [57], see also [79], while behaviors of two subspaces in generic position are well understood from all aspects; see, e.g., [3, 6, 25, 26, 42, 54, 55, 57, 66, 68, 71, 79, 80, 98, 100, 111, 118, 148]. The concept of commuting subspaces was dened in [93, 112]. The JDSDIs in Theorems 2.5(a) and 2.6(a) enable us to suciently characterize behaviors of two subspaces and their algebraic operations, such as, disjoints of subspaces, commutativity of spaces, space inequalities and space identities are some simple relations between subspaces. In what follows, we collect some elementary or well-known results in the literature on disjoints of subspaces, commutativity of spaces, space inequalities and space identities, and present some new conclusions on these algebraic properties.

m Corollary 2.13. Let M and N be two linear subspaces of C . Then, the following statements are equivalent: h1i M ∩ N = {0}, namely, M and N are disjoint. ⊥ ⊥ m h2i M + N = C . h3i M + N = M ⊕ N. h4i M = M ◦ N⊥ and/or N = N ◦ M⊥. h5i M + N = (M ◦ N⊥) ⊕ (N ◦ M⊥). h6i M ⊆ M⊥ + N⊥ and/or N ⊆ M⊥ + N⊥. h7i M = (M ∩ N⊥) ⊕ (M • N) and/or N = (M⊥ ∩ N) ⊕ (N • M). h8i M + N ⊆ M⊥ + N⊥. h9i M + N = (M ∩ N⊥) ⊕ (M⊥ ∩ N) ⊕ (M • N) ⊕ (N • M). h10i M + N⊥ = (M ∩ N⊥) ⊕ (M⊥ ∩ N⊥) ⊕ (M • N) ⊕ (N⊥ • M). h11i M⊥ + N = (M⊥ ∩ N) ⊕ (M⊥ ∩ N⊥) ⊕ (M⊥ • N) ⊕ (N • M). ⊥ ⊥ ⊥ ⊥ m ⊥ h12i Comm(M, N) = (M ∩ N ) ⊕ (M ∩ N) ⊕ (M ∩ N ) with some/all of the four facts: C = (M ∩ N ) ⊕ ⊥ ⊥ ⊥ m ⊥ ⊥ ⊥ ⊥ ⊥ (M ∩N)⊕(M ∩N )⊕(M•N)⊕(N•M), C = (M∩N )⊕(M ∩N)⊕(M ∩N )⊕(M•N)⊕(N •M), m ⊥ ⊥ ⊥ ⊥ ⊥ m ⊥ ⊥ ⊥ C = (M ∩ N ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊕ (M • N) ⊕ (N • M), C = (M ∩ N ) ⊕ (M ∩ N) ⊕ (M ∩ N⊥) ⊕ (M⊥ • N) ⊕ (N⊥ • M). h13i PM = PM∩N⊥ + PM•N and/or PN = PM⊥∩N + PN•M. h14i PMPN = PM•NPN•M and/or PNPM = PN•MPM•N. h15i dim(M + N) = dim(M) + dim(N). h16i dim(M⊥ + N⊥) = m. h17i dim(M⊥ ∩ N⊥) = m − dim(M) − dim(N). h18i dim(M) = dim(M ◦ N⊥) and/or dim(N) = dim(N ◦ M⊥). h19i dim(M + N) = dim(M ◦ N⊥) + dim(N ◦ M⊥). h20i dim(M) = dim(M ∩ N⊥) + dim(M • N) and/or dim(N) = dim(M⊥ ∩ N) + dim(N • M). h21i Some/all of the three facts: dim(M + N) = dim(M ∩ N⊥) + dim(M⊥ ∩ N) + dim(M • N) + dim(N • M) and some/all dim(M + N⊥) = dim(M ∩ N⊥) + dim(M⊥ ∩ N⊥) + dim(M • N) + dim(N⊥ • M), dim(M⊥ + N) = dim(M⊥ ∩ N) + dim(M⊥ ∩ N⊥) + dim(M⊥ • N) + dim(N • M). h22i dim(M + N) = dim(M + N⊥) − dim(M⊥ + N) + 2 dim(N) = dim(M⊥ + N) − dim(M + N⊥) + 2 dim(M). h23i dim(M + N) = 2 dim(M) + dim(M⊥ ∩ N) − dim(M ∩ N⊥) = 2 dim(N) + dim(M ∩ N⊥) − dim(M⊥ ∩ N). h24i dim(M ∩ N⊥) + dim(M⊥ ∩ N) + dim(M⊥ ∩ N⊥) + dim(M • N) + dim(N • M) = m. 154 Ë Yongge Tian h25i dim(M ∩ N⊥) + dim(M⊥ ∩ N) + dim(M⊥ ∩ N⊥) + dim(M • N) + dim(N⊥ • M) = dim(M ∩ N⊥) + dim(M⊥ ∩ N) + dim(M⊥ ∩ N⊥) + dim(M⊥ • N) + dim(N • M) = dim(M ∩ N⊥) + dim(M⊥ ∩ N) + dim(M⊥ ∩ N⊥) + dim(M⊥ • N) + dim(N⊥ • M) = m.

Other three groups of equivalent statement can be established similarly for M ∩ N⊥ = {0}, M⊥ ∩ N = {0}, and M⊥ ∩ N⊥ = {0} to hold. Combining Theorem 2.13 with the three groups of equivalent statements will yield a family of identifying conditions for two subspaces to be in generic position. We shall present some of them in Theorem 4.2.

m Corollary 2.14. Let M and N be two linear subspaces of C . Then, the following statements are equivalent: h1i M and N are commutative, i.e., M = (M ∩ N) ⊕ (M ∩ N⊥) and/or N = (M ∩ N) ⊕ (M⊥ ∩ N). h2i M and N⊥ are commutative, i.e., M = (M ∩ N) ⊕ (M ∩ N⊥) and/or N⊥ = (M ∩ N⊥) ⊕ (M⊥ ∩ N⊥). h3i M⊥ and N are commutative, i.e., M⊥ = (M⊥ ∩ N) ⊕ (M⊥ ∩ N⊥) and/or N = (M ∩ N) ⊕ (M⊥ ∩ N). h4i M⊥ and N⊥ are commutative, i.e., M⊥ = (M⊥ ∩ N) ⊕ (M⊥ ∩ N⊥) and/or N⊥ = (M ∩ N⊥) ⊕ (M⊥ ∩ N⊥). h5i Some/all of the four facts: M = (M+N)∩(M+N⊥), N = (M+N)∩(M⊥ +N), M⊥ = (M⊥ +N)∩(M⊥ +N⊥)), N⊥ = (M + N⊥) ∩ (M⊥ + N⊥). h6i Some/all of the eight facts: M ⊇ N ◦ M and/or M ⊇ N⊥ ◦ M, N ⊇ M ◦ N, N ⊇ M⊥ ◦ N, M⊥ ⊇ N ◦ M⊥, M⊥ ⊇ N⊥ ◦ M⊥, N⊥ ⊇ M ◦ N⊥, N⊥ ⊇ M⊥ ◦ N⊥. h7i Some/all of the eight facts: M ⊆ (M∩N)⊕N⊥, M ⊆ (M∩N⊥)⊕N, N ⊆ (M∩N)⊕M⊥, N ⊆ (M⊥ ∩N)⊕M, M⊥ ⊆ (M⊥ ∩ N) ⊕ N⊥, M⊥ ⊆ (M⊥ ∩ N⊥) ⊕ N, N⊥ ⊆ (M ∩ N⊥) ⊕ M⊥, N⊥ ⊆ (M⊥ ∩ N⊥) ⊕ M. h8i Some/all of the four facts: M∩N = M◦N = N◦M, M∩N⊥ = M◦N⊥ = N⊥◦M, M⊥∩N = M⊥◦N = N◦M⊥, M⊥ ∩ N⊥ = M⊥ ◦ N⊥ = N⊥ ◦ M⊥. h9i Some/all of the four facts: M+N = M⊕(M⊥∩N) = N⊕(M∩N⊥), M+N⊥ = M⊕(M⊥∩N⊥) = N⊥⊕(M∩N), M⊥ + N = M⊥ ⊕ (M ∩ N) = N ⊕ (M⊥ ∩ N⊥), M⊥ + N⊥ = M⊥ ⊕ (M ∩ N⊥) = N⊥ ⊕ (M⊥ ∩ N). h10i Some/all of the four facts: M ◦ N = N ◦ M, M ◦ N⊥ = N⊥ ◦ M, M⊥ ◦ N = N ◦ M⊥, M⊥ ◦ N⊥ = N⊥ ◦ M⊥. h11i Some/all of the four facts: M ⊕ (M⊥ ∩ N) = N ⊕ (M ∩ N⊥) and/or M ⊕ (M⊥ ∩ N⊥) = N⊥ ⊕ (M ∩ N), M⊥ ⊕ (M ∩ N⊥) = N⊥ ⊕ (M⊥ ∩ N), M⊥ ⊕ (M ∩ N) = N ⊕ (M⊥ ∩ N⊥). h12i Some/all of the four facts: M + N = (M ∩ N) ⊕ (M ∩ N⊥) ⊕ (M⊥ ∩ N), M + N⊥ = (M ∩ N) ⊕ (M ∩ N⊥) ⊕ (M⊥ ∩N⊥), M⊥ +N = (M∩N)⊕(M⊥ ∩N)⊕(M⊥ ∩N⊥), M⊥ +N⊥ = (M∩N⊥)⊕(M⊥ ∩N)⊕(M⊥ ∩N⊥). m h13i Comm(M, N) = C . h14i Some/all of the four facts: M⊥ ∩ N⊥ = (M + N⊥) ∩ (M⊥ + N) ∩ (M⊥ + N⊥) and/or M⊥ ∩ N = (M⊥ + N⊥) ∩ (M⊥ +N)∩(M+N), M∩N⊥ = (M+N)∩(M+N⊥)∩(M⊥ +N⊥), M∩N = (M+N)∩(M+N⊥)∩(M⊥ +N). h15i Some/all of the four facts: M • N = (M ◦ N) ∩ (M ◦ N⊥) = {0}, N • M = (N ◦ M) ∩ (N ◦ M⊥) = {0}, M⊥ • N = (M⊥ ◦ N) ∩ (M⊥ ◦ N⊥) = {0}, N⊥ • M = (N⊥ ◦ M) ∩ (N⊥ ◦ M⊥) = {0}. m ⊥ ⊥ ⊥ m ⊥ ⊥ ⊥ h16i Some/all of the four facts: C = M ⊕ (M ∩ N) ⊕ (M ∩ N ), C = N ⊕ (M ∩ N ) ⊕ (M ∩ N ), m ⊥ ⊥ m ⊥ ⊥ C = M ⊕ (M ∩ N) ⊕ (M ∩ N ), C = N ⊕ (M ∩ N) ⊕ (M ∩ N). h17i Some/all of the four facts: PM = PM∩N + PM∩N⊥ , PN = PM∩N + PM⊥∩N, PM⊥ = PM⊥∩N + PM⊥∩N⊥ , PN⊥ = PM∩N⊥ + PM⊥∩N⊥ . h18i PM + PN = 2PM∩N + PM∩N⊥ + PM⊥∩N and/or PM⊥ + PN⊥ = PM⊥∩N + PM∩N⊥ + 2PM⊥∩N⊥ . h19i PM+N = PM∩N + PM∩N⊥ + PM⊥∩N and/or PM⊥+N⊥ = PM∩N⊥ + PM⊥∩N + PM⊥∩N⊥ . h20i Im = PM∩N + PM∩N⊥ + PM⊥∩N + PM⊥∩N⊥ . h21i Some/all of the four facts: dim(M) = dim(M ∩ N) + dim(M ∩ N⊥), dim(N) = dim(M ∩ N) + dim(M⊥ ∩ N), dim(M⊥) = dim(M⊥ ∩ N) + dim(M⊥ ∩ N⊥), dim(N⊥) = dim(M ∩ N⊥) + dim(M⊥ ∩ N⊥). h22i Some/all of the four facts: dim(M + N) = dim(M ∩ N) + dim(M ∩ N⊥) + dim(M⊥ ∩ N), dim(M + N⊥) = dim(M∩N)+dim(M∩N⊥)+dim(M⊥ ∩N⊥), dim(M⊥ +N) = dim(M∩N)+dim(M⊥ ∩N)+dim(M⊥ ∩N⊥), dim(M⊥ + N⊥) = dim(M ∩ N⊥) + dim(M⊥ ∩ N) + dim(M⊥ ∩ N⊥). h23i Some/all of the four facts: dim(M ∩ N) = dim(M ◦ N) = dim(N ◦ M), dim(M ∩ N⊥) = dim(M ◦ N⊥) = dim(N⊥ ◦ M), dim(M⊥ ∩ N) = dim(M⊥ ◦ N) = dim(N ◦ M⊥), dim(M⊥ ∩ N⊥) = dim(M⊥ ◦ N⊥) = dim(N⊥ ◦ M⊥). Linear subspaces and orthogonal projectors Ë 155 h24i Some/all of the four facts: dim(M + N) = dim(M) + dim(M⊥ ∩ N) = dim(N) + dim(M ∩ N⊥), dim(M + N⊥) = dim(M) + dim(M⊥ ∩ N⊥) = dim(N⊥) + dim(M ∩ N), dim(M⊥ + N) = dim(M⊥) + dim(M ∩ N) = dim(N) + dim(M⊥ ∩ N⊥), dim(M⊥ + N⊥) = dim(M⊥) + dim(M ∩ N⊥) = dim(N⊥) + dim(M⊥ ∩ N).

More equivalent statements for two subspaces to be commutative will be given in Theorem 5.1.

m Corollary 2.15. Let M and N be two linear subspaces of C . Then, the following statements are equivalent: h1i M ⊇ N and/or M⊥ ⊆ N⊥. h2i M + N = M and/or M⊥ ∩ N⊥ = M⊥. h3i M⊥ + N⊥ = N⊥ and/or M ∩ N = N. h4i M = (M ∩ N⊥) ⊕ N and/or M⊥ = (M⊥ + N) ∩ N⊥. h5i N⊥ = M⊥ ⊕ (M ∩ N⊥) and/or N = M ∩ (M⊥ + N). ⊥ ⊥ ⊥ m ⊥ ⊥ ⊥ h6i (M ∩ N) ⊕ (M ∩ N ) ⊕ (M ∩ N ) = C and/or (M + N) ∩ (M + N) ∩ (M + N ) = {0}. h7i dim( M + N ) = dim(M) and/or dim( M⊥ + N⊥ ) = dim(N⊥). h8i dim(M ∩ N) = dim(N) and/or dim(M⊥ ∩ N⊥) = dim(M⊥). h9i dim(M ∩ N) + dim(M ∩ N⊥) + dim(M⊥ ∩ N⊥) = m. h10i M⊥ ∩ N = {0} with M and N being commutative. h11i M⊥ ∩ N = {0} with one/both of the two facts: N = (M + N) ∩ (M⊥ + N), M⊥ = (M⊥ + N) ∩ (M⊥ + N⊥). h12i M⊥ ∩ N = {0} with some/all of the seven facts: M ⊇ N⊥ ◦ M, N ⊇ M ◦ N, N ⊇ M⊥ ◦ N, M⊥ ⊇ N ◦ M⊥, M⊥ ⊇ N⊥ ◦ M⊥, N⊥ ⊇ M ◦ N⊥, N⊥ ⊇ M⊥ ◦ N⊥. h13i M⊥ ∩N = {0} with some/all of the six facts: M ⊆ (M∩N)⊕N⊥, M ⊆ (M∩N⊥)⊕N, N ⊆ (M∩N)⊕M⊥, M⊥ ⊆ (M⊥ ∩ N⊥) ⊕ N, N⊥ ⊆ (M ∩ N⊥) ⊕ M⊥, N⊥ ⊆ (M⊥ ∩ N⊥) ⊕ M. h14i M⊥ ∩ N = {0} with some/all of the three facts: M ◦ N = M ∩ N, M ◦ N⊥ = M ∩ N⊥, N⊥ ◦ M = M ∩ N⊥. h15i M⊥ ◦ N = {0} and/or N ◦ M⊥ = {0}. h16i M⊥ ∩ N = {0} with one/both of the two facts: M⊥ ◦ N⊥ = M⊥ ∩ N⊥, N⊥ ◦ M⊥ = M⊥ ∩ N⊥. h17i M⊥ ∩ N = {0} and M + N = N ⊕ (M ∩ N⊥). ⊥ ⊥ m ⊥ m h18i M ⊕ (M ∩ N ) = C and/or N ⊕ (M ∩ N) = C . h19i M⊥ ∩ N = {0} with one/both of the two facts: M⊥ ⊕ N = M⊥ ⊕ (M ∩ N), M⊥ ⊕ N = N ⊕ (M⊥ ∩ N⊥). h20i M⊥ ∩ N = {0} and M⊥ + N⊥ = M⊥ ⊕ (M ∩ N⊥). h21i M⊥ ∩ N = {0} with one/both of the two facts: M ◦ N = N, M⊥ ◦ N⊥ = M⊥. h22i M⊥ ∩ N = {0} with one/both of the two facts: M ◦ N⊥ = N⊥ ◦ M, M⊥ ◦ N = N ◦ M⊥. h23i M⊥ ∩ N = {0} with one/both of the two facts: M ⊕ (M⊥ ∩ N⊥) = N⊥ ⊕ (M ∩ N), M⊥ ⊕ (M ∩ N) = N ⊕ (M⊥ ∩ N⊥). h24i M⊥∩N = {0} with some/all of the three facts: M+N = (M∩N)⊕(M∩N⊥), M⊥+N = (M∩N)⊕(M⊥∩N⊥), M⊥ + N⊥ = (M ∩ N⊥) ⊕ (M⊥ ∩ N⊥). h25i M⊥ ∩ N = {0} with one/both of the two facts: M⊥ ∩ N⊥ = (M⊥ + N) ∩ (M⊥ + N⊥), M ∩ N⊥ = (M + N) ∩ (M⊥ + N⊥), M ∩ N = (M + N) ∩ (M⊥ + N). h26i M⊥ ∩ N = {0} with one/both of the two facts: M • N = {0}, N⊥ • M = {0}. ⊥ ⊥ ⊥ m ⊥ ⊥ ⊥ h27i M ∩N = {0} with one/both of the two facts: M +(M∩N)+(M∩N ) = C , N+(M∩N )+(M ∩N ) = m C . ⊥ h28i M ∩ N = {0} with one/both of the two facts: PM = PM∩N + PM∩N⊥ , PN⊥ = PM∩N⊥ + PM⊥∩N⊥ . ⊥ h29i M ∩ N = {0} with one/both of the two facts: PN = PM∩N, PM⊥ = PM⊥∩N⊥ . ⊥ h30i M ∩ N = {0} with one/both of the two facts: PM + PN = 2PM∩N + PM∩N⊥ , PM⊥ + PN⊥ = PM∩N⊥ + 2PM⊥∩N⊥ . ⊥ h31i M ∩ N = {0} with one/both of the two facts: PM+N = PM∩N + PM∩N⊥ , PM⊥+N⊥ = PM∩N⊥ + PM⊥∩N⊥ . ⊥ h32i M ∩ N = {0} and Im = PM∩N + PM∩N⊥ + PM⊥∩N⊥ . h33i M⊥ ∩ N = {0} with one/both of the two facts: dim(M) = dim(M ∩ N) + dim(M ∩ N⊥), dim(N⊥) = dim(M ∩ N⊥) + dim(M⊥ ∩ N⊥). h34i M⊥ ∩N = {0} with some/all of the three facts: dim(M+N) = dim(M∩N)+dim(M∩N⊥), dim(M⊥ +N) = dim(M ∩ N) + dim(M⊥ ∩ N⊥), dim(M⊥ + N⊥) = dim(M ∩ N⊥) + dim(M⊥ ∩ N⊥). h35i M⊥ ∩ N = {0} and dim(M + N⊥) = dim(M ∩ N) + dim(M ∩ N⊥) + dim(M⊥ ∩ N⊥). 156 Ë Yongge Tian h36i M⊥ ∩ N = {0} with some/all of the ve facts: dim(M ◦ N) = dim(M ∩ N), dim(M ◦ N⊥) = dim(M ∩ N⊥), dim(N⊥ ◦ M) = dim(M ∩ N⊥), dim(M⊥ ◦ N⊥) = dim(M⊥ ∩ N⊥), dim(N⊥ ◦ M⊥) = dim(M⊥ ∩ N⊥). h37i M⊥ ∩ N = {0} with some/all of the four facts: dim(M + N) = dim(N) + dim(M ∩ N⊥), dim(M⊥ + N) = dim(M⊥) + dim(M ∩ N), dim(M⊥ + N) = dim(N) + dim(M⊥ ∩ N⊥), dim(M⊥ + N⊥) = dim(M⊥) + dim(M ∩ N⊥).

More equivalent statements for two subspaces to satisfy inequalities will be given in Theorem 6.1.

m Corollary 2.16. Let M and N be two linear subspaces of C . Then, the following statements are equivalent: h1i M = N, i.e., both M ⊇ N and M ⊆ N. h2i M⊥ = N⊥, i.e., both M⊥ ⊇ N⊥ and M⊥ ⊆ N⊥. h3i {M ⊆ N and dim(M) = dim(N)} and/or {M ⊇ N and dim(M) = dim(N)}. h4i {M⊥ ⊆ N⊥ and dim(M⊥) = dim(N⊥)} and/or {M⊥ ⊇ N⊥ and dim(M⊥) = dim(N⊥)}. h5i M + N = M ∩ N and/or M⊥ + N⊥ = M⊥ ∩ N⊥. h6i {M = N ⊕ (M ∩ N⊥) and N = M ⊕ (M⊥ ∩ N)} and/or {M⊥ = N⊥ ⊕ (M⊥ ∩ N) and N⊥ = M⊥ ⊕ (M ∩ N⊥)}. h7i {M = N ∩ (M + N⊥) and N = M ∩ (M⊥ + N)} and/or {M⊥ = N⊥ ∩ (M⊥ + N) and N⊥ = M⊥ ∩ (M + N⊥)}. ⊥ ⊥ m ⊥ ⊥ h8i (M ∩ N) ⊕ (M ∩ N ) = C and/or (M + N) ∩ (M + N ) = {0}. h9i 2 dim( M + N ) = dim(M) + dim(N) and/or 2 dim( M⊥ + N⊥ ) = dim(M⊥) + dim(N⊥). h10i dim( M + N ) = dim(M) = dim(N) and/or dim( M⊥ + N⊥ ) = dim(M⊥) = dim(N⊥). h11i dim(M + N) = dim(M ∩ N) and/or dim( M⊥ + N⊥ ) = dim(M⊥ ∩ N⊥). h12i dim(M ∩ N) = dim(M) = dim(N) and/or dim(M⊥ ∩ N⊥) = dim(M⊥) = dim(N⊥). h13i dim(M ∩ N) + dim(M⊥ ∩ N⊥) = m. h14i dim(M + N) + dim(M⊥ + N⊥) = m. h15i Some/all of the four facts: {M ∩ N⊥ = {0} and M ⊇ N}, {M⊥ ∩ N = {0} and M ⊆ N}, {M ∩ N⊥ = {0} and M⊥ ⊆ N⊥}, {M⊥ ∩ N = {0} and M⊥ ⊇ N⊥}. h16i M ∩ N⊥ = M⊥ ∩ N = {0}, and M and N are commutative. h17i M ∩ N⊥ = M⊥ ∩ N = {0} with some/all of the four facts: M = M ∩ N, N = M ∩ N, M⊥ = M⊥ ∩ N⊥, N⊥ = M⊥ ∩ N⊥. h18i M ∩ N⊥ = M⊥ ∩ N = {0} with some/all of the four facts: M = M + N, N = M + N, M⊥ = M⊥ + N⊥, N⊥ = M⊥ + N⊥. h19i M ∩ N⊥ = M⊥ ∩ N = {0} with some/all of the four facts: M ⊇ N, N ⊇ M, M⊥ ⊇ N⊥, N⊥ ⊇ M⊥. h20i M ∩ N⊥ = M⊥ ∩ N = {0} with some/all of the four facts: M ◦ N⊥ = {0}, N ◦ M⊥ = {0}, M⊥ ◦ N = {0}, N⊥ ◦ M = {0}. ⊥ ⊥ ⊥ ⊥ m ⊥ ⊥ m h21i M ∩ N = M ∩ N = {0} with some/all of the four facts: M ⊕ (M ∩ N ) = C , N ⊕ (M ∩ N ) = C , ⊥ m ⊥ m M ⊕ (M ∩ N) = C , N ⊕ (M ∩ N) = C . h22i M∩N⊥ = M⊥∩N = {0} with some/all of the four facts: M◦N = N◦M, M◦N⊥ = N⊥◦M, M⊥◦N = N◦M⊥, M⊥ ◦ N⊥ = N⊥ ◦ M⊥. h23i M∩N⊥ = M⊥ ∩N = {0} with one/both of the two facts: M⊕(M⊥ ∩N⊥) = N⊥ ⊕(M∩N), M⊥ ⊕(M∩N) = N ⊕ (M⊥ ∩ N⊥). h24i M ∩ N⊥ = M⊥ ∩ N = {0} with some/all of the four facts: M • N = {0}, N • M = {0}, M⊥ • N = {0}, N⊥ • M = {0}. ⊥ ⊥ h25i M ∩ N = M ∩ N = {0} with some/all of the four facts: PM = PM∩N, PN = PM∩N, PM⊥ = PM⊥∩N⊥ , PN⊥ = PM⊥∩N⊥ . ⊥ ⊥ h26i M ∩ N = M ∩ N = {0} with one/both of the two facts: PM + PN = 2PM∩N, PM⊥ + PN⊥ = 2PM⊥∩N⊥ . ⊥ ⊥ h27i M ∩ N = M ∩ N = {0} with one/both of the two facts: PM+N = PM∩N, PM⊥+N⊥ = PM⊥∩N⊥ . ⊥ ⊥ h28i M ∩ N = M ∩ N = {0} and Im = PM∩N + PM⊥∩N⊥ . h29i M ∩ N⊥ = M⊥ ∩ N = {0} with some/all of the four facts: dim(M) = dim(M ∩ N), dim(N) = dim(M ∩ N), dim(M⊥) = dim(M⊥ ∩ N⊥), dim(N⊥) = dim(M⊥ ∩ N⊥). h30i M ∩ N⊥ = M⊥ ∩ N = {0} with some/all of the four facts: dim(M + N) = dim(M), dim(M + N) = dim(N), dim(M⊥ + N⊥) = dim(M⊥), dim(M⊥ + N⊥) = dim(N⊥). Linear subspaces and orthogonal projectors Ë 157 h31i M ∩ N⊥ = M⊥ ∩ N = {0} with some/all of the four facts: dim(M) + dim(M⊥ ∩ N⊥) = m, dim(N) + dim(M⊥ ∩ N⊥) = m, dim(M⊥) + dim(M ∩ N) = m, dim(N⊥) + dim(M ∩ N) = m.

More equivalent statements for two subspaces to be equal will be given in Theorem 6.2.

m Corollary 2.17. Let M and N be two linear subspaces of C . Then, the following statements are equivalent: h1i M ⊥ N, namely, M and N are orthogonal. h2i M ⊆ N⊥ and/or M⊥ ⊇ N. h3i M⊥ + N = M⊥ and/or M + N⊥ = N⊥. h4i M ∩ N⊥ = M and/or M⊥ ∩ N = N. h5i M = (M + N) ∩ N⊥ and/or N = M⊥ ∩ (M + N). h6i M⊥ = (M⊥ ∩ N⊥) ⊕ N and/or N⊥ = M ⊕ (M⊥ ∩ N⊥). ⊥ ⊥ ⊥ ⊥ m ⊥ ⊥ h7i (M ∩ N) ⊕ (M ∩ N ) ⊕ (M ∩ N ) = C and/or (M + N) ∩ (M + N) ∩ (M + N ) = {0}. h8i dim( M⊥ + N ) = dim(M⊥) and/or dim( M + N⊥ ) = dim(N⊥). h9i dim(M⊥ ∩ N) = dim(N) and/or dim(M ∩ N⊥) = dim(M). h10i dim(M⊥ ∩ N) + dim(M⊥ ∩ N⊥) + dim(M ∩ N⊥) = m. h11i M ∩ N = {0} with some/all of the three facts: M⊥ and N are commutative, M⊥ and N⊥ are commutative, M and N⊥ are commutative. h12i M ∩ N = {0} with one/both of the two facts: N = (M⊥ + N) ∩ (M + N), M = (M + N) ∩ (M + N⊥). h13i M∩N = {0} with some/all of the eight facts: M ⊇ N ◦M, M ⊇ N⊥ ◦M, M⊥ ⊇ N ◦M⊥, M⊥ ⊇ N⊥ ◦M⊥, N ⊇ M⊥ ◦ N, N ⊇ M ◦ N, N⊥ ⊇ M⊥ ◦ N⊥, N⊥ ⊇ M ◦ N⊥. h14i M∩N = {0} with some/all of the six facts: M ⊆ (M∩N⊥)⊕N, N ⊆ (M⊥ ∩N)⊕M, M⊥ ⊆ (M⊥ ∩N)⊕N⊥, M⊥ ⊆ (M⊥ ∩ N⊥) ⊕ N, N⊥ ⊆ (M⊥ ∩ N⊥) ⊕ M, N⊥ ⊆ (M ∩ N⊥) ⊕ M⊥. h15i M∩N = {0} with some/all of the three facts: M⊥◦N = M⊥∩N, M⊥◦N⊥ = M⊥∩N⊥, N⊥◦M⊥ = M⊥∩N⊥. h16i M ◦ N = {0} and/or N ◦ M = {0}. h17i M ∩ N = {0} with one/both of the two facts: M ◦ N⊥ = M ∩ N⊥, N⊥ ◦ M = M ∩ N⊥. h18i M ∩ N = {0} with one/both of the two facts: M⊥ + N = N ⊕ (M⊥ ∩ N⊥), M + N⊥ = M ⊕ (M⊥ ∩ N⊥). ⊥ ⊥ m ⊥ ⊥ m h19i M ⊕ (M ∩ N ) = C and/or N ⊕ (M ∩ N) = C . h20i M ∩ N = {0} with one/both of the two facts: M ⊕ N = M ⊕ (M⊥ ∩ N), M ⊕ N = N ⊕ (M ∩ N⊥). h21i M ∩ N = {0} with one/both of the two facts: M + N⊥ = M ⊕ (M⊥ ∩ N⊥), N + M⊥ = N ⊕ (M⊥ ∩ N⊥). h22i M ∩ N = {0} with one/both of the two facts: M⊥ ◦ N = N, M ◦ N⊥ = M. h23i M ∩ N = {0} with one/both of the two facts: M ◦ N = N ◦ M, M⊥ ◦ N⊥ = N⊥ ◦ M⊥. h24i M∩N = {0} with one/both of the two facts: M⊥⊕(M∩N⊥) = N⊥⊕(M⊥∩N), M⊕(M⊥∩N) = N⊕(M∩N⊥). h25i M ∩ N = {0} with some/all of the three facts: M + N = (M⊥ ∩ N) ⊕ (M ∩ N⊥), M⊥ + N = (M⊥ ∩ N) ⊕ (M⊥ ∩ N⊥), M + N⊥ = (M⊥ ∩ N⊥) ⊕ (M ∩ N⊥). h26i M∩N = {0} with some/all of the three facts: M∩N⊥ = (M+N)∩(M+N⊥), M⊥ ∩N = (M⊥ +N)∩(M+N), M⊥ ∩ N⊥ = (M⊥ + N) ∩ (M + N⊥). h27i M ∩ N = {0} with one/both of the two facts: M⊥ • N = {0}, N⊥ • M⊥ = {0}. ⊥ ⊥ ⊥ m ⊥ ⊥ ⊥ h28i M∩N = {0} with one/both of the two facts: M+(M ∩N)+(M ∩N ) = C , N+(M ∩N )+(M∩N ) = m C . h29i M ∩ N = {0} with one/both of the two facts: PM⊥ = PM⊥∩N + PM⊥∩N⊥ , PN = PM⊥∩N. h30i M ∩ N = {0} with one/both of the two facts: PM = PM∩N⊥ , PN⊥ = PM⊥∩N⊥ + PM∩N⊥ . h31i M ∩ N = {0} with one/both of the two facts: PM + PN⊥ = PM⊥∩N⊥ + 2PM∩N⊥ , PM⊥ + PN = 2PM⊥∩N + PM⊥∩N⊥ . h32i M∩N = {0} with one/both of the two facts: PM+N⊥ = PM⊥∩N⊥ +PM∩N⊥ , PM⊥+N = PM⊥∩N +PM⊥∩N⊥ . h33i M ∩ N = {0} and Im = PM⊥∩N + PM⊥∩N⊥ + PM∩N⊥ . h34i M ∩ N = {0} with one/both of the two facts: dim(M⊥) = dim(M⊥ ∩ N) + dim(M⊥ ∩ N⊥), dim(N⊥) = dim(M⊥ ∩ N⊥) + dim(M ∩ N⊥). h35i M ∩ N = {0} and dim(M⊥ + N) = dim(M⊥ ∩ N) + dim(M⊥ ∩ N⊥). h36i M ∩ N = {0} and dim(M + N) = dim(M⊥ ∩ N) + dim(M ∩ N⊥). h37i M ∩ N = {0} and dim(M + N⊥) = dim(M⊥ ∩ N⊥) + dim(M ∩ N⊥). 158 Ë Yongge Tian h38i M ∩ N = {0} and dim(M⊥ + N⊥) = dim(M⊥ ∩ N) + dim(M⊥ ∩ N⊥) + dim(M ∩ N⊥). h39i M∩N = {0} and with some/all of the ve facts: dim(M⊥ ◦N) = dim(M⊥ ∩N), dim(M⊥ ◦N⊥) = dim(M⊥ ∩ N⊥), dim(N⊥ ◦ M⊥) = dim(M⊥ ∩ N⊥), dim(M ◦ N⊥) = dim(M ∩ N⊥), dim(N⊥ ◦ M) = dim(M ∩ N⊥). h40i M ∩ N = {0} with some/all of the four facts: dim(M⊥ + N) = dim(N) + dim(M⊥ ∩ N⊥), dim(M + N) = dim(M)+dim(M⊥ ∩ N), dim(M+N) = dim(N)+dim(M ∩ N⊥), dim(M+N⊥) = dim(M)+dim(M⊥ ∩ N⊥).

Descriptions of a sequence of linear subspaces are very complicated problems in linear algebra. Some previous considerations on three or more linear subspaces and their joint positions were scattered in the literature associated with pure algebra and Hilbert space; see e.g., [47, 66, 68, 70, 94, 95, 108, 139].

3 The simultaneous CS decomposition of two orthogonal projectors and their consequences

m Let M and N be two linear subspaces of C . An essential characterization of the relative position of the two subspaces is to dene canonical angles between the vectors in them, which measure how the two subspaces dier in a natural way. Many results related to canonical angles between two subspaces can be found in the literature; see e.g., [31, 37, 42, 53, 55, 56, 71, 72, 106, 107, 124, 150, 151]. In particular, canonical angles between two subspaces can be used to construct simultaneous decomposition identities of the two projectors onto the two subspaces. During this approach, various matrix decompositions, including the well-known CS decomposition, can be used; see, e.g., [27, 28, 42, 71, 74, 79, 86, 114, 130, 150]. In this section, we reconsider the CS decomposition for two orthogonal projectors, and use the decomposition with precise sizes of submatrices to characterize many behaviors of the pair of orthogonal projectors and their operations.

m Lemma 3.1. Let M and N be two linear subspaces of C , and dene

A = PM, B = PN, (3.1)

Ae = PM⊥ = Im − PM = Im − A, Be = PN⊥ = Im − PN = Im − B. (3.2)

Then, there exists a unitary matrix U such that

∗ A = Udiag Ik1 , 0k2 , Ik3 , Ik4 , 0k5 , 0k6 U , (3.3) " # ! C2 CS B = Udiag , I , 0 , I , 0 U∗, (3.4) CSS2 k3 k4 k5 k6 ∗ Ae = Udiag 0k1 , Ik2 , 0k3+k4 , Ik5+k6 U , (3.5) " 2 # ! S −CS ∗ Be = Udiag , 0 , I , 0 , I U , (3.6) −CSC2 k3 k4 k5 k6 where C and S are positive diagonal real matrices such that

2 2 C + S = Ik1 , (3.7) and the sizes of the submatrices in (3.3)–(3.6) are given by

k1 = k2 = r[ A, B ] − r(A) − r(B) + r(AB), k3 = r(A) + r(B) − r[ A, B ], (3.8)

k4 = r(A) − r(AB), k5 = r(B) − r(AB), k6 = m − r[ A, B ], (3.9)

0 6 k1 6 m/2, 0 6 k3, k4, k5, k6 6 m. (3.10)

A special case of Lemma 3.1 is given below. Linear subspaces and orthogonal projectors Ë 159

m×m Corollary 3.2. Let M ∈ C . Then, there exists a unitary matrix U such that † ∗ MM = Udiag Ik1 , 0k2 , Ik3+k4 , 0k5+k6 U , (3.11) " # ! C2 CS M†M = Udiag , I , 0 , I , 0 U∗, (3.12) CSS2 k3 k4 k5 k6 † ∗ Im − MM = Udiag 0k1 , Ik2 , 0k3+k4 , Ik5+k6 U , (3.13) " 2 # ! † S −CS ∗ Im − M M = Udiag , 0 , I , 0 , I U , (3.14) −CSC2 k3 k4 k5 k6

2 2 where C and S are positive diagonal real matrices such that C + S = Ik1 , the sizes of the blocks in (3.11)–(3.14) are given by

∗ 2 ∗ k1 = k2 = r[ M, M ] − 2r(M) + r(M ), k3 = 2r(M) − r[ M, M ], (3.15) 2 ∗ k4 = k5 = r(M) − r(M ), k6 = m − r[ M, M ]. (3.16)

In particular, under the condition r(M2) = r(M), (3.11)–(3.16) reduce to " # ! C2 CS MM† = Udiag I , 0 , I , 0 U∗, M†M = Udiag , I , 0 U∗, k1 k2 k3 k6 CSS2 k3 k6

" 2 # ! † ∗ † S −CS ∗ Im − MM = Udiag 0 , I , 0 , I U , Im − M M = Udiag , 0 , I U , k1 k2 k3 k6 −CSC2 k3 k6

∗ ∗ ∗ where k1 = k2 = r[ M, M ] − r(M), k3 = 2r(M) − r[ M, M ] and k6 = m − r[ M, M ]. Under the condition M2 = 0, (3.11)–(3.16) reduce to

† ∗ † ∗ MM = Udiag Ik4 , 0k5+k6 U , M M = Udiag 0k4 , Ik5 , 0k6 U , † ∗ † ∗ Im − MM = Udiag 0k4 , Ik5+k6 U , Im − M M = Udiag Ik4 , 0k5 , Ik6 U , where k4 = k5 = r(M) and k6 = m − 2r(M).

Lemma 3.3. Any Hermitian matrix M can uniquely be decomposed as M = M+ + M−, where M+ and M−, called the positive semi-denite and negative semi-denite parts of M, satisfy

M+ < 0, M− 4 0, M+M− = M−M+ = 0. (3.17) In this situation, the following results hold. 1 2 1/2 1 2 1/2 2 1/2 (a) M+ = 2 [(M ) + M], M− = 2 [M − (M ) ], and M+ − M− = (M ) . k k 2k 2k−1 2k−1 (b) (M )+ = (M+) , (M )− = 0, and (M )− = (M−) . k k k (c) M = (M+) + (M−) . † † † † (d) (M+) = (M )+ and (M−) = (M )−. † † † (e) M = M+ + M−. (f) i+(M) = r(M+), i−(M) = r(M−), and r(M+) + r(M−) = r(M). (g) R(M+) ⊆ R(M), R(M−) ⊆ R(M), and R(M+) ⊕ R(M−) = R(M).

Proof. It follows directly from the spectral decomposition of Hermitian matrix. A most valuable part of Lemma 3.1 is giving the analytical formulas for calculating the sizes of all submatrices in (3.3), which enable us to precisely characterize many algebraic properties of the two orthogonal projectors and their operations. In what follows, we give a variety of simultaneous decomposition identities of A ± B, AB, BA, AB±BA, ABA, BAB, ABA±BAB, as well as R(A)∩R(B), R(A)∩R(Be), R(Ae)∩R(B) and R(Ae)∩R(Be), which allow us to characterize behaviors of orthogonal projectors and their operations in many cases. It can be seen in particular that the diagonal elements of C2 are the singular values of the product of AB (eigenvalues of the product of ABA) less than 1. 160 Ë Yongge Tian

m Theorem 3.4. Let M and N be two linear subspaces of C , and assume that A, B, Ae and Be in (3.2) are decomposed as (3.3)–(3.6), respectively. Also denote Cb = Ik1 + C and bS = Ik1 + S. Then, the following results hold. (a) There exists a unitary matrix U such that the following results hold simultaneously " # ! I + C2 CS A + B = Udiag k1 , 2I , I , 0 U∗, CSS2 k3 k4+k5 k6 " 1/2 2 −1/2 −1/2# √ ! 1/2 1 bS + C bS CSbS ∗ (A + B) = Udiag √ , 2Ik , Ik +k , 0k U , 2 CSbS−1/2 SbS1/2 3 4 5 6 " 2 # ! Ik1 + S −CS ∗ Im + A − B = Udiag , I , 2I , 0 , I U , −CSC2 k3 k4 k5 k6 " 1/2 2 −1/2 −1/2# √ ! 1/2 1 Cb + S Cb −CSCb ∗ (Im + A − B) = Udiag √ , Ik , 2Ik , 0k , Ik U , 2 −CSCb−1/2 CCb1/2 3 4 5 6 " 2 # ! C CS ∗ Im − A + B = Udiag 2 , Ik3 , 0k4 , 2Ik5 , Ik6 U , CSIk1 + S " 1/2 −1/2 # √ ! 1/2 1 CCb CSCb ∗ (Im − A + B) = Udiag √ , Ik , 0k , 2Ik , Ik U , 2 CSCb−1/2 Cb1/2 + S2Cb−1/2 3 4 5 6 " 2 # ! −C −CS ∗ Im − A − B = Udiag , −I , 0 , I U , −CSC2 k3 k4+k5 k6

2 2 2 ∗ ( Im − A − B ) = Udiag C , C , Ik3 , 0k4+k5 , Ik6 U , 2 1/2 ∗ [( Im − A − B ) ] = Udiag C, C, Ik3 , 0k4+k5 , Ik6 U , " # ! S2 −CS A − B = Udiag , 0 , I , −I , 0 U∗, −CS −S2 k3 k4 k5 k6

2 2 2 ∗ ( A − B ) = Udiag S , S , 0k3 , Ik4+k5 , 0k6 U , 2 1/2 ∗ [( A − B ) ] = Udiag S, S, 0k3 , Ik4+k5 , 0k6 U , " 2 # ! S −CS ∗ 2Im − A − B = Udiag 2 , 0k3 , Ik4+k5 , 2Ik6 U , −CSIk1 + C " 1/2 −1/2 # √ ! 1/2 1 SbS −CSbS ∗ (2Im − A − B) = Udiag √ , 0k , Ik +k , 2Ik U , 2 −CSbS−1/2 bS1/2 + C2bS−1/2 3 4 5 6 " # ! " # ! C2 CS CS AB = Udiag , I , 0 U∗, (AB)1/2 = Udiag , I , 0 U∗, 0 0 k3 k4+k5+k6 0 0 k3 k4+k5+k6 " # ! " # ! C2 0 C 0 BA = Udiag , I , 0 U∗, (BA)1/2 = Udiag , I , 0 U∗, CS 0 k3 k4+k5+k6 S 0 k3 k4+k5+k6 " # ! I 0 A + B − AB = Udiag k1 , I , 0 U∗, CSS2 k3+k4+k5 k6 " # ! I 0 (A + B − AB)1/2 = Udiag k1 , I , 0 U∗, CSSˆ−1 S k3+k4+k5 k6 " # ! I CS A + B − BA = Udiag k1 , I , 0 U∗, 0 S2 k3+k4+k5 k6 " # ! I CSSˆ−1 (A + B − BA)1/2 = Udiag k1 , I , 0 U∗, 0 S k3+k4+k5 k6 Linear subspaces and orthogonal projectors Ë 161

" # ! 2C2 CS AB + BA = Udiag , 2I , 0 U∗, CS 0 k3 k4+k5+k6

" 2 # ! 2S −CS ∗ 2Im − AB − BA = Udiag , 0k3 , 2Ik4+k5+k6 U , −CS 2Ik1 −1 −1 2 4 Im + AB + BA = ( 2 Im − A − B ) " −1 2 # ! 4 Im + 2C CS −1 −1 ∗ = Udiag −1 , (2 + 4 )Ik3 , 4 Ik4+k5+k6 U , CS 4 Im " # ! 0 CS AB − BA = Udiag , 0 U∗, −CS 0 m−2k1

2 2 2 2 2 ∗ ( AB − BA ) = −Udiag C S , C S , 0m−2k1 U , √ 2 1/2 ∗ [( AB − BA ) ] = −1Udiag CS, CS, 0m−2k1 U , " 2 # ! S −CS ∗ Im − AB = Udiag , 0k3 , Ik4+k5+k6 U , 0 Ik1 " −1# ! 1/2 S −CSSˆ ∗ (Im − AB) = Udiag , 0k3 , Ik4+k5+k6 U , 0 Ik1 " 2 # ! S 0 ∗ Im − BA = Udiag , 0k3 , Ik4+k5+k6 U , −CSIk1 " # ! 1/2 S 0 ∗ (Im − BA) = Udiag , 0 , I U , ˆ−1 k3 k4+k5+k6 −CSS Ik1 " 2 # ! S −CS ∗ ABe = Udiag , 0 , I , 0 U , 0 0 k3 k4 k5+k6 " # ! 1/2 S −C ∗ (ABe) = Udiag , 0 , I , 0 U , 0 0 k3 k4 k5+k6

" 2 # ! S 0 ∗ BAe = Udiag , 0 , I , 0 U , −CS 0 k3 k4 k5+k6 " # ! 1/2 S 0 ∗ (BAe ) = Udiag , 0 , I , 0 U , −C 0 k3 k4 k5+k6 " # ! 0 CS ∗ BAe = Udiag , 0 , I , 0 U , 0 S2 k3+k4 k5 k6 " # ! 1/2 0 C ∗ (BAe) = Udiag , 0 , I , 0 U , 0 S k3+k4 k5 k6 " # ! 0 0 ∗ ABe = Udiag , 0 , I , 0 U , CSS2 k3+k4 k5 k6 " # ! 1/2 0 0 ∗ (ABe ) = Udiag , 0 , I , 0 U , CS k3+k4 k5 k6 " # ! 0 0 ∗ AeBe = Udiag , 0 , I U , −CSC2 k3+k4+k5 k6 " # ! 1/2 0 0 ∗ (AeBe) = Udiag , 0 , I U , −SC k3+k4+k5 k6 162 Ë Yongge Tian

" # ! 0 −CS ∗ BeAe = Udiag , 0 , I U , 0 C2 k3+k4+k5 k6 " # ! 1/2 0 −S ∗ (BeAe) = Udiag , 0 , I U , 0 C k3+k4+k5 k6

" 2 # ! 2S −CS ∗ ABe + BAe = Udiag , 0 , 2I , 0 U , −CS 0 k3 k4 k5+k6 " # ! 0 CS ∗ ABe + BAe = Udiag , 0 , 2I , 0 U , CS 2S2 k3+k4 k5 k6 " # ! 0 −CS ∗ AeBe + BeAe = Udiag , 0 , 2I U , −CS 2C2 k3+k4+k5 k6

2 ∗ ABA = Udiag C , 0k2 , Ik3 , 0k4+k5+k6 U , 1/2 ∗ (ABA) = Udiag C, 0k2 , Ik3 , 0k4+k5+k6 U , " # ! C4 C3S BAB = Udiag , I , 0 U∗, C3SC2S2 k3 k4+k5+k6 " # ! C3 C2S (BAB)1/2 = Udiag , I , 0 U∗, C2SCS2 k3 k4+k5+k6 " # ! 0 CS ∗ ABAe = −ABeAe = Udiag , 0 U , 0 0 m−2k1 " # ! 0 0 ∗ ABAe = −AeBAe = Udiag , 0 U , CS 0 m−2k1

" 2 2 3 # ! C S −C S ∗ BABe = −BAeBe = Udiag , 0 U , CS3 −C2S2 m−2k1

" 2 2 3 # ! C S CS ∗ BABe = −BeABe = Udiag , 0 U , −C3S −C2S2 m−2k1

2 ∗ ABAe = Udiag S , 0k2+k3 , Ik4 , 0k5+k6 U , 1/2 ∗ (ABAe ) = Udiag S, 0k2+k3 , Ik4 , 0k5+k6 U , " 2 2 3# ! C S CS ∗ BABe = Udiag , 0 U , CS3 S4 m−2k1

" 2 2# ! 1/2 C SCS ∗ (BABe ) = Udiag , 0 U , CS2 S3 m−2k1

2 ∗ ABe Ae = Udiag 0k1 , S , 0k3+k4 , Ik5 , 0k6 U , 1/2 ∗ (ABe Ae) = Udiag 0k1 , S, 0k3+k4 , Ik5 , 0k6 U , " 4 3# ! S −CS ∗ BAe Be = Udiag , 0 , I , 0 U , −CS3 C2S2 k3 k4 k5+k6

" 3 2# ! 1/2 S −CS ∗ (BAe Be) = Udiag , 0 , I , 0 U , −CS2 C2S k3 k4 k5+k6

2 ∗ AeBeAe = Udiag 0k1 , C , 0k3+k4+k5 , Ik6 U , 1/2 ∗ (AeBeAe) = Udiag 0k1 , C, 0k3+k4+k5 , Ik6 U , Linear subspaces and orthogonal projectors Ë 163

" 2 2 3 # ! C S −C S ∗ BeAeBe = Udiag , 0 , I U , −C3SC4 k3+k4+k5 k6

" 2 2 # ! 1/2 CS −C S ∗ (BeAeBe) = Udiag , 0 , I U , −C2SC3 k3+k4+k5 k6 " # ! C4 + C2 C3S ABA + BAB = Udiag , 2I , 0 U∗, C3SC2S2 k3 k4+k5+k6 " # ! C2S2 −C3S ABA − BAB = Udiag , 0 U∗, −C3S −C2S2 m−2k1

2 4 2 4 2 ∗ ( ABA − BAB ) = Udiag C S , C S , 0m−2k1 U ,

2 1/2 2 2 ∗ [( ABA − BAB ) ] = Udiag C S, C S, 0m−2k1 U ,

2 ∗ Im − ABA = Udiag S , Ik2 , 0k3 , Ik4+k5+k6 U ,

" 2 2 2 3 # ! S + C S −C S ∗ Im − BAB = Udiag 3 2 2 , 0k3 , Ik4+k5+k6 U , −C SIk1 − C S " 2 2 2 3 # ! 2S + C S −C S ∗ 2Im − ABA − BAB = Udiag 3 2 2 , 0k3 , 2Ik4+k5+k6 U , −C S 2Ik2 − C S " # ! S2 + C2S2 −C3S A − BAB = Udiag , 0 , I , 0 U∗, −C3S −C2S2 k3 k4 k5+k6 " # ! 0 CS B − ABA = Udiag , 0 , I , 0 U∗, CSS2 k3+k4 k5 k6

2 2 ∗ ABA + ABe Ae = Udiag C , S , Ik3 , 0k4 , Ik5 , 0k6 U , " # ! 0 CS ∗ ABAe + ABAe = Udiag , 0 U , CS 0 m−2k1

" 4 4 3 3# ! C + S C S − CS ∗ BAB + BAe Be = Udiag , I , 0 U , C3S − CS3 2C2S2 k3+k4 k5+k6

" 2 2 3 3 # ! 2C S CS − C S ∗ BABe + BABe = Udiag , 0 U . CS3 − C3S 2S2C2 m−2k1

(b) There exists a unitary matrix U such that the following results hold simultaneously

" 2 # ! 1 S + S −CS ∗ (A − B)+ = Udiag , 0 , I , 0 U , 2 −CSS − S2 k3 k4 k5+k6

" 2 # ! 1 S − S −CS ∗ ( A − B)− = Udiag , 0 , −I , 0 U , 2 −CS −S2 − S k3+k4 k5 k6

" 2 # ! 1 C + C CS ∗ (A + B − Im)+ = Udiag , I , 0 U , 2 CSC − C2 k3 k4+k5+k6

" 2 # ! 1 C − CCS ∗ (A + B − Im)− = Udiag , 0 , −I U , 2 CS −C2 − C k3+k4+k5 k6

" 2 # ! 1 CCb SCCb ∗ (AB + BA)+ = Udiag , 2Ik , 0k +k +k U , 2 SCCCSb 2 3 4 5 6 " 2 # ! 1 −C(Ik1 − C) SC(Ik1 − C) ∗ (AB + BA)− = Udiag , 0 U , 2 2 m−2k1 SC(Ik1 − C) −CS 164 Ë Yongge Tian

" 2 # ! 1 SbS −SCbS ∗ (ABe + BAe )+ = Udiag , 0k , 2Ik , 0k +k U , 2 −SCbSC2S 3 4 5 6 " 2 # ! 1 −S(Ik1 − S) −SC(Ik1 − S) ∗ (ABe + BAe )− = Udiag , 0 U , 2 2 m−2k1 −SC(Ik1 − S) −C S " 2 # ! 1 C SSCbS ∗ (ABe + BAe)+ = Udiag , 0k +k , 2Ik , 0k U , 2 SCbSSbS2 3 4 5 6 " 2 # ! 1 −C SSC(Ik1 − S) ∗ (ABe + BAe)− = Udiag , 0 U , 2 2 m−2k1 SC(Ik1 − S) −S(Ik1 − S) " 2 # ! 1 CS −SCCb ∗ (AeBe + BeAe)+ = Udiag , 0k +k +k , 2Ik U , 2 −SCCCb Cb2 3 4 5 6 " 2 # ! 1 −CS −SC(Ik1 − C) ∗ (AeBe + BeAe)− = Udiag , 0 U , 2 2 m−2k1 −SC(Ik1 − C) −C(Ik1 − C) " 2 3 # ! 1 C SbS −C S ∗ (ABA − BAB)+ = Udiag , 0 U , 2 3 2 m−2k1 −C SC S(Ik1 − S) " 2 3 # ! 1 −C S(Ik1 − S) −C S ∗ (ABA − BAB)− = Udiag , 0m−2k U . 2 −C3S −C2SbS 1

r(AB) = r(BA) = r(ABA) = r(BAB) = k1 + k3,

r[ A, B ] = r(A + B) = r(A + B − AB) = r(A + B − BA) = 2k1 + k3 + k4 + k5,

r(A + B − Im) = 2k1 + k3 + k6, r(A − B) = r[ ABe, BAe ] = r[ BAe , ABe ] = r[ ABAe , BABe ] = r[ BAe Be, ABe Ae ] = r[ A − BAB, B − ABA ] = r(ABAe + BABe ) = r(BAe Be + ABe Ae)

= 2k1 + k4 + k5,

r[ Im − A, Im − B ] = r(2Im − A − B) = r(Im − AB) = r(Im − BA)

= r(Im − ABA) = r(Im − BAB) = 2k1 + k4 + k5 + k6,

r[ AB, BA ] = r[ ABA, BAB ] = r(AB + BA) = r(ABA + BAB) = 2k1 + k3,

r[ Im − AB, Im − BA ] = r[ Im − ABA, Im − BAB ] = r(2Im − AB − BA)

= r(2Im − ABA − BAB) = m − k3,

r(AB − BA) = r(ABA − BAB) = r(ABAe + ABAe ) = r(BABe + BABe ) = 2k1,

r(ABe) = r(BAe ) = r(ABAe ) = r(BAe Be) = k1 + k4,

r(BAe) = r(ABe ) = r(BABe ) = r(ABe Ae) = k1 + k5,

r(AeBeAe) = r(BeAeBe) = k1 + k6, r[ABe, BAe ] = r[ABAe , BAe Be] = r(ABe + BAe ) = r(ABAe + BAe Be) = r(A − BAB)

= 2k1 + k4, r[ABe , BAe] = r[ABe Ae, BABe ] = r(ABe + BAe) = r(ABe Ae + BABe ) = r(B − ABA)

= 2k1 + k5,

r[AeBe, BeAe] = r[AeBeAe, BeAeBe] = r(AeBe + BeAe) = r(AeBeAe + BeAeBe) = 2k1 + k6,

r(ABAe) = r(ABeAe) = r(ABAe ) = r(AeBAe ) = r(BABe) = r(BAeBe) = k1,

r(ABA + ABe Ae) = 2k1 + k3 + k5,

r(BAB + BAe Be) = 2k1 + k3 + k4, Linear subspaces and orthogonal projectors Ë 165

r[(A − B)+] = k1 + k4, r[( A − B)−] = k1 + k5,

r[(A + B − Im)+] = k1 + k3, r[(A + B − Im)−] = k1 + k4,

r[(AB + BA)+] = k1 + k3, r[(AB + BA)−] = k1,

r[(ABe + BAe )+] = k1 + k4, r[(ABe + BAe )−] = k1,

r[(ABe + BAe)+] = k1 + k5, r[(ABe + BAe)−] = k1,

r[(AeBe + BeAe)+] = k1 + k6, r[(AeBe + BeAe)−] = k1,

r[(ABA − BAB)+] = r[(ABA − BAB)−] = k1.

(d) There exists a unitary matrix U such that

" −1 # ! † Ik1 −CS −1 ∗ ( A + B ) = Udiag −1 −2 2 , 2 Ik3 , Ik4+k5 , 0k6 U , −CS S ( Ik1 + C ) " −1 # ! † Ik1 C S ∗ ( A + B − Im ) = Udiag −1 , Ik3 , 0k4+k5 , −Ik6 U , C S −Ik1 " −1# ! † Ik1 −CS ∗ ( A − B ) = Udiag −1 , 0k3 , Ik4 , −Ik5 , 0k6 U , −CS −Ik1 " −2 2 −1# ! † S ( Ik1 + C ) CS −1 ∗ ( 2Im − A − B ) = Udiag −1 , 0k3 , Ik4+k5 , 2 Ik6 U , CS Ik1 " # ! I 0 (AB)† = Udiag k1 , I , 0 U∗, C−1S 0 k3 k4+k5+k6 " # ! I C−1S (BA)† = Udiag k1 , I , 0 U∗, 0 0 k3 k4+k5+k6 " # ! I 0 ( A + B − AB )† = Udiag k1 , I , 0 U∗, −CS−1 S−2 k3+k4+k5 k6 " # ! I −CS−1 ( A + B − BA )† = Udiag k1 , I , 0 U∗, 0 S−2 k3+k4+k5 k6 " # ! 0 (CS)−1 ( AB + BA )† = Udiag , 2−1I , 0 U∗, (CS)−1 −2S−2 k3 k4+k5+k6 " # ! 0 −(CS)−1 ( AB − BA )† = Udiag , 0 U∗, (CS)−1 0 m−2k1

" −2 −1# ! † S CS ∗ ( Im − AB ) = Udiag , 0k3 , Ik4+k5+k6 U , 0 Ik1 " −2 # ! † S 0 ∗ ( Im − BA ) = Udiag −1 , 0k3 , Ik4+k5+k6 U , CS Ik1 " # ! † I 0 ∗ (ABe) = Udiag k1 , 0 , I , 0 U , −CS−1 0 k3 k4 k5+k6

" −1# ! † I −CS ∗ (BAe ) = Udiag k1 , 0 , I , 0 U , 0 0 k3 k4 k5+k6 " # ! † 0 0 ∗ (BAe) = Udiag −1 , 0k3+k4 , Ik5 , 0k6 U , CS Ik1 " −1# ! † 0 CS ∗ (ABe ) = Udiag , 0k3+k4 , Ik5 , 0k6 U , 0 Ik1 166 Ë Yongge Tian

" −1 # ! † 0 −C S ∗ (AeBe) = Udiag , 0k3+k4+k5 , Ik6 U , 0 Ik1 " # ! † 0 0 ∗ (BeAe) = Udiag −1 , 0k3+k4+k5 , Ik6 U , −C SIk1 " −1# ! † 0 −(CS) −1 ∗ (ABe + BAe ) = Udiag , 0 , 2 I , 0 U , −(CS)−1 −2C−2 k3 k4 k5+k6

" −2 −1# ! † −2C (CS) −1 ∗ (ABe + BAe) = Udiag , 0 , 2 I , 0 U , (CS)−1 0 k3+k4 k5 k6

" −2 −1# ! † −2S −(CS) −1 ∗ (AeBe + BeAe) = Udiag , 0 , 2 I U , −(CS)−1 0 k3+k4+k5 k6

† −2 ∗ (ABA) = Udiag C , 0k2 , Ik3 , 0k4+k5+k6 U , " # ! I C−1S (BAB)† = Udiag k1 , I , 0 U∗, C−1SC−2S2 k3 k4+k5+k6 " # ! † † 0 0 ∗ (ABAe) = −(ABeAe) = Udiag , 0 U , (CS)−1 0 m−2k1

" −1# ! † † 0 (CS) ∗ (ABAe ) = −(AeBAe ) = Udiag , 0 U , 0 0 m−2k1

" −1 # ! † † Ik1 C S ∗ (BABe) = −(BAeBe) = Udiag −1 , 0m−2k1 U , −CS −Ik1 " −1# ! † † Ik1 −CS ∗ (BABe ) = −(BeABe ) = Udiag −1 , 0m−2k1 U , C S −Ik1 † −2 ∗ (ABAe ) = Udiag S , 0k2+k3 , Ik4 , 0k5+k6 U ,

" 2 −2 −1# ! † C S CS ∗ (BABe ) = Udiag −1 , 0m−2k1 U , CS Ik1 † −2 ∗ (ABe Ae) = Udiag 0k1 , S , 0k3+k4 , Ik5 , 0k6 U ,

" −1# ! † I −CS ∗ (BAe Be) = Udiag k1 , 0 , I , 0 U , −CS−1 C2S−2 k3 k4 k5+k6

† −2 ∗ (AeBeAe) = Udiag 0k1 , C , 0k3+k4+k5 , Ik6 U ,

" −2 2 −1 # ! † C S −C S ∗ (BeAeBe) = Udiag −1 , 0k3+k4+k5 , Ik6 U , −C SIk1 " # ! C−2 −(CS)−1 ( ABA + BAB )† = Udiag , 2−1I , 0 U∗, −(CS)−1 S−2 + (CS)−2 k3 k4+k5+k6 " # ! C−2 −(CS)−1 ( ABA − BAB )† = Udiag , 0 U∗, −(CS)−1 −C−2 m−2k1

† −2 ∗ ( Im − ABA ) = Udiag S , Ik1 , 0k3 , Ik4+k5+k6 U ,

" −2 2 3 −1 # ! † S − C C S ∗ ( Im − BAB ) = Udiag 3 −1 2 , 0k3 , Ik4+k5+k6 U , C S Ik1 + C  −1  " 2 2 2 3 # † 2S + C S −C S −1 ∗ (2Im − ABA − BAB) = Udiag 3 2 2 , 0k3 , 2 Ik4+k5+k6U , −C S 2Ik2 − C S Linear subspaces and orthogonal projectors Ë 167

" −1 # ! † Ik1 −CS ∗ ( A − BAB ) = Udiag −1 −2 , 0k3 , Ik4 , 0k5+k6 U , −CS −Ik1 − C " # ! −C−2 (CS)−1 ( B − ABA )† = Udiag , 0 , I , 0 U∗, (CS)−1 0 k3+k4 k5 k6

† −2 −2 ∗ ( ABA + ABe Ae ) = Udiag C , S , Ik3 , 0k4 , Ik5 , 0k6 U ,

" −1# ! † 0 (CS) ∗ ( ABAe + ABAe ) = Udiag , 0 U , (CS)−1 0 m−2k1

" −1 −1 # ! † 2I C S − CS ∗ ( BAB + BAe Be ) = Udiag k1 , I , 0 U , C−1S − CS−1 C2S−2 + C−2S2 k3+k4 k5+k6

" −1 −1# ! † 2Ik1 C S − CS ∗ ( BABe + BABe ) = Udiag −1 −1 , 0m−2k1 U , C S − CS −2Ik1 " # ! 1 I + S−1 −CS−1 (A − B)† = Udiag k1 , 0 , I , 0 U∗, + 2 −1 −1 k3 k4 k5+k6 −CS S − Ik1 " # ! 1 I − S−1 −CS−1 ( A − B)† = Udiag k1 , 0 , −I , 0 U∗, − 2 −1 −1 k3+k4 k5 k6 −CS −Ik1 − S " −1 −1 # ! † 1 Ik1 + C C S ∗ (A + B − Im) = Udiag , I , 0 U , + 2 −1 −1 k3 k4+k5+k6 C SC − Ik1 " −1 −1 # ! † 1 Ik1 − C C S ∗ (A + B − Im) = Udiag , 0 , −I U , − 2 −1 −1 k3+k4+k5 k6 C S −Ik1 − C " # ! 1 C−1 (CS)−1(I − C) (AB + BA)† = Udiag k1 , 2−1I , 0 U∗, + 2 −1 −1 −2 2 k3 k4+k5+k6 (CS) (Ik1 − C) C S (Ik1 − C) " −1 −1 # ! † 1 −C (CS) Cb ∗ (AB + BA)− = Udiag , 0k +k +k +k U , 2 (CS)−1Cb −C−1S−2Cb2 3 4 5 6 " −1 −1 # ! † 1 S −(CS) (Ik1 − S) −1 ∗ (ABe + BAe )+ = Udiag , 0 , 2 I , 0 U , 2 −1 −2 −1 2 k3 k4 k5+k6 −(CS) (Ik1 − S) C S (Ik1 − S) " −1 −1 # ! † 1 −S −(CS) bS ∗ (ABe + BAe )− = Udiag , 0m−2k U , 2 −(CS)−1bS −C−2S−1bS2 1 " −2 −1 2 −1 # ! † 1 C S (Ik1 − S) (CS) (Ik1 − S) −1 ∗ (ABe + BAe)+ = Udiag , 0 , 2 I , 0 U , 2 −1 −1 k3+k4 k5 k6 (CS) (Ik1 − S) S " −2 −1 2 −1 # ! † 1 −C S (Ik1 + S) (CS) (Ik1 − S) ∗ (ABe + BAe)− = Udiag , 0 U , 2 −1 −1 m−2k1 (CS) (Ik1 − S) −S " −1 −2 2 −1 # ! † 1 C S (Ik1 − C) −(CS) (Ik1 − C) −1 ∗ (AeBe + BeAe)+ = Udiag , 0 , 2 I U , 2 −1 −1 k3+k4+k5 k6 −(CS) (Ik1 − C) C " −1 −2 2 −1 # ! † 1 −C S (Ik1 + C) −(CS) Cb ∗ (AeBe + BeAe)− = Udiag , 0m−2k U , 2 −(CS)−1Cb −C−1 1 " −2 −1 −1 # ! 1 C S bS −(CS) ∗ (ABA − BAB)† = Udiag , 0 U , + 2 −1 −2 −1 m−2k1 −(CS) C S (Ik1 − S) " −2 −1 −1 # ! † 1 −C S (Ik1 − S) −(CS) ∗ (ABA − BAB)− = Udiag , 0m−2k U , 2 −(CS)−1 −C−2S−1bS 1 † ∗ (A + B)(A + B) = Udiag(I2k1 , 0m−2k1 )U , 168 Ë Yongge Tian

† ∗ (Im + A − B)(Im + A − B) = Udiag(I2k1+k3+k4 , 0k5 , Ik6 )U , † ∗ (Im − A − B)(Im − A − B) = Udiag(I2k1+k3 , 0k4+k5 , Ik6 )U , † ∗ (A − B)(A − B) = Udiag(I2k1 0k3 , Ik4+k5 , 0k6 )U , † ∗ (2Im − A − B)(2Im − A − B) = Udiag(I2k1 , 0k3 , Ik4+k5+k6 )U , † † ∗ (AB)(AB) = (ABA)(ABA) = Udiag(Ik1 , 0k2 , Ik3 , 0k4+k5+k6 )U , " # ! C2 CS (BA)(BA)† = (BAB)(BAB)† = Udiag , I , 0 U∗, CSS2 k3 k4+k5+k6 † ∗ (A + B − AB)(A + B − AB) = Udiag(Im−k6 , 0k6 )U , † ∗ (A + B − BA)(A + B − BA) = Udiag(Im−k6 , 0k6 )U , † ∗ (AB + BA)(AB + BA) = Udiag(I2k1+k3 , 0k4+k5+k6 )U , † ∗ (2Im − AB − BA)(2Im − AB − BA) = Udiag(I2k1 , 0k3 , Ik4+k5+k6 )U , † ∗ (AB − BA)(AB − BA) = Udiag(I2k1 , 0m−2k1 )U , † ∗ (Im − AB)(Im − AB) = Udiag(I2k1 , 0k3 , Ik4+k5+k6 )U , † ∗ (Im − BA)(Im − BA) = Udiag(I2k1 , 0k3 , Ik4+k5+k6 )U , † † ∗ (ABe)(ABe) = (ABAe )(ABAe ) = Udiag(Ik1 , 0k2+k3 , Ik4 , 0k5+k6 )U , " 2 # ! † † S −CS ∗ (BAe )(BAe ) = (BAe Be)(BAe Be) = Udiag , 0 , I , 0 U , −CSC2 k3 k4 k5+k6

" 2 # ! † † C CS ∗ (BAe)(BAe) = (BABe )(BABe ) = Udiag , 0 , I , 0 U , CSS2 k3+k4 k5 k6 † † ∗ (ABe )(ABe ) = (ABe Ae)(ABe Ae) = Udiag(0k1 , Ik2 , 0k3+k4 , Ik5 , 0k6 )U , † † ∗ (AeBe)(AeBe) = (AeBeAe)(AeBeAe) = Udiag(0k1 , Ik2 , 0k3+k4+k5 , Ik6 )U , " 2 # ! † † S −CS ∗ (BeAe)(BeAe) = (BeAeBe)(BeAeBe) = Udiag , 0 , I U , −CSC2 k3+k4+k5 k6 † ∗ (ABe + BAe )(ABe + BAe ) = Udiag(I2k1 , 0k3 , Ik4 , 0k5+k6 )U , † ∗ (ABe + BAe)(ABe + BAe) = Udiag(I2k1 , 0k3+k4 , Ik5 , 0k6 )U , † ∗ (AeBe + BeAe)(AeBe + BeAe) = Udiag( I2k1 , 0k3+k4+k5 , Ik6 )U , † † ∗ (ABAe)(ABAe) = (ABeAe)(ABeAe) = Udiag(Ik1 , 0m−k1 )U , † † ∗ (ABAe )(ABAe ) = (AeBAe )(AeBAe ) = Udiag(0k1 , Ik1 , 0m−2k1 )U , " 2 # ! † † C CS ∗ (BABe)(BABe) = (BAeBe)(BAeBe) = Udiag , 0 U , CSS2 m−2k1

" 2 # ! † † S −CS ∗ (BABe )(BABe ) = (BeABe )(BeABe ) = Udiag , 0 U , −CSC2 m−2k1 † ∗ (ABA + BAB)(ABA + BAB) = Udiag( I2k1+k3 , 0k4+k5+k6 )U , † ∗ (ABA − BAB)(ABA − BAB) = Udiag( I2k1 , 0m−2k1 )U , † ∗ (Im − ABA)(Im − ABA) = Udiag( I2k1 , 0k3 , Ik4+k5+k6 )U , † ∗ (Im − BAB)(Im − BAB) = Udiag( I2k1 , 0k3 , Ik4+k5+k6 )U , † ∗ ( 2Im − ABA − BAB )( 2Im − ABA − BAB ) = Udiag( I2k1 , 0k3 , Ik4+k5+k6 )U , † ∗ (A − BAB)(A − BAB) = Udiag(I2k1 , 0k3 , Ik4 , 0k5+k6 )U , † ∗ (B − ABA)(B − ABA) = Udiag(I2k1 , 0k3+k4 , Ik5 , 0k6 )U , Linear subspaces and orthogonal projectors Ë 169

† ∗ (ABA + ABe Ae)(ABA + ABe Ae) = Udiag( I2k1+k3 , 0k4 , Ik5 , 0k6 )U , † ∗ (ABAe + ABAe )(ABAe + ABAe ) = Udiag( I2k1 , 0m−2k1 )U , † ∗ (BAB + BAe Be)(BAB + BAe Be) = Udiag( I2k1+k3+k4 , 0k5+k6 )U , † ∗ (BABe + BABe )(BABe + BABe ) = Udiag( I2k1 , 0m−2k1 )U , " # ! C−2 C−3S (AB)# = Udiag , I , 0 U∗, 0 0 k3 m−2k1−k3 " # ! C−2 0 (BA)# = Udiag , I , 0 U∗, C−3S 0 k3 m−2k1−k3 " # ! I C−1S (AB)(AB)# = Udiag k1 , I , 0 U∗, 0 0 k3 m−2k1−k3 " # ! I 0 (BA)(BA)# = Udiag k1 , I , 0 U∗, C−1S 0 k3 m−2k1−k3 " # ! 2C−2 C−3S (AB)# + (BA)# = Udiag , 2I , 0 U∗. C−3S 0 k3 m−2k1−k3

(e) The orthogonal projectors onto M∩N, M∩N⊥, M⊥ ∩N, and M⊥ ∩N⊥ can simultaneously be decomposed as

∗ ∗ ⊥ PM∩N = Udiag 02k1 , Ik3 , 0k4+k5+k6 U , PM∩N = Udiag 02k1+k3 , Ik4 , 0k5+k6 U , ∗ ∗ ⊥ ⊥ ⊥ PM ∩N = Udiag 02k1+k3+k4 , Ik5 , 0k6 U , PM ∩N = Udiag 0m−k6 , Ik6 U ,

and the dimensions of M ∩ N, M ∩ N⊥, M⊥ ∩ N, and M⊥ ∩ N⊥ are

dim(M ∩ N) = r(A) + r(B) − r[ A, B ], dim(M ∩ N⊥) = r(A) − r(AB), dim(M⊥ ∩ N) = r(B) − r(AB), dim(M⊥ ∩ N⊥) = m − r[ A, B ].

(f) The following results hold

AB = BA ⇔ ABA = BAB ⇔ ABAe + ABAe = 0 ⇔ BABe + BABe = 0

⇔ r[ A, B ] = r(A) + r(B) − r(AB) ⇔ k1 = k2 = 0,

M ∩ N = {0} ⇔ r[ A, B ] = r(A) + r(B) ⇔ k3 = 0, ⊥ M ∩ N = {0} ⇔ r(A) = r(AB) ⇔ k4 = 0, ⊥ M ∩ N = {0} ⇔ r(B) = r(AB) ⇔ k5 = 0, m ⊥ ⊥ M + N = C ⇔ M ∩ N = {0} ⇔ r[ A, B ] = m ⇔ k6 = 0.

(g) The following equivalent statements hold

⊥ M ⊆ N ⇔ BA = A ⇔ M ∩ N = {0} and AB = BA ⇔ k1 = k2 = k4 = 0, ⊥ M ⊇ N ⇔ AB = B ⇔ M ∩ N = {0} and AB = BA ⇔ k1 = k2 = k5 = 0, ⊥ ⊥ M = N ⇔ A = B ⇔ M ∩ N = M ∩ N = {0} and AB = BA ⇔ k1 = k2 = k4 = k5 = 0, M ⊥ N ⇔ AB = 0 ⇔ AB + BA = 0 ⇔ ABA = 0 ⇔ BAB = 0 ⇔ ABA + BAB = 0

⇔ k1 = k2 = k3 = 0,

A − B is nonsigular ⇔ k3 = k6 = 0,

AB + BA is nonsigular ⇔ ABA + BAB is nonsigular ⇔ k4 = k5 = k6 = 0,

AB − BA is nonsigular ⇔ ABA − BAB is nonsigular ⇔ k3 = k4 = k5 = k6 = 0,

Im − AB is nonsigular ⇔ Im − ABA is nonsigular ⇔ Im − BAB is nonsigular ⇔ k3 = 0, 170 Ë Yongge Tian

A − BAB is nonsigular ⇔ k3 = k5 = k6 = 0,

B − ABA is nonsigular ⇔ k3 = k4 = k6 = 0.

Proof. It follows from direct verications by (3.3)–(3.10) and the denitions of the Moore–Penrose inverses and group inverses of matrices.

It is easy to observe from Theorem 3.4(a) that

R(AB) = M ◦ N = M ∩ (M ∩ N⊥)⊥, R(ABe) = M ◦ N⊥ = M ∩ (M ∩ N)⊥, R(ABe ) = M⊥ ◦ N = M⊥ ∩ (M⊥ ∩ N⊥)⊥, R(AeBe) = M⊥ ◦ N⊥ = M⊥ ∩ (M⊥ ∩ N)⊥, R(BA) = N ◦ M = N ∩ (M⊥ ∩ N)⊥, R(BAe ) = N⊥ ◦ M = N⊥ ∩ (M⊥ ∩ N⊥)⊥, R(BAe) = N ◦ M⊥ = N ∩ (M ∩ N)⊥, R(BeAe) = N⊥ ◦ M⊥ = N⊥ ∩ (M ∩ N⊥)⊥.

These identities can be used to establish many direct sum decomposition identities of A, B, Ae, and Be in (3.2) and their algebraic operations. In particular, we have the following consequences.

m Corollary 3.5. Let M and N be two linear subspaces of C , and assume that A, B, Ae and Be in (3.2) are decomposed as (3.3)–(3.6). Then, the following results hold. (a) The following intersection identities hold

R(ABAe) = R(ABeAe) = R(AB) ∩ R(ABe) = M • N, R(BABe) = R(BAeBe) = R(BA) ∩ R(BAe) = N • M, ⊥ R(ABAe ) = R(AeBAe ) = R(ABe ) ∩ R(AeBe) = M • N, ⊥ R(BABe ) = R(BeABe ) = R(BAe ) ∩ R(BeAe) = N • M. (b) The following direct sum decomposition identities hold

R(AB) = R(ABA) = R(ABAe) ⊕ (M ∩ N) = (M • N) ⊕ (M ∩ N), ⊥ ⊥ R(ABe) = R(ABAe ) = R(ABAe) ⊕ (M ∩ N ) = (M • N) ⊕ (M ∩ N ), ⊥ ⊥ ⊥ R(ABe ) = R(ABe Ae) = R(ABAe ) ⊕ (M ∩ N) = (M • N) ⊕ (M ∩ N), ⊥ ⊥ ⊥ ⊥ ⊥ R(AeBe) = R(AeBeAe) = R(ABAe ) ⊕ (M ∩ N ) = (M • N) ⊕ (M ∩ N ), R(BA) = R(BAB) = R(BABe) ⊕ (M ∩ N) = (N • M) ⊕ (M ∩ N), ⊥ ⊥ R(BAe) = R(BABe ) = R(BABe) ⊕ (M ∩ N) = (N • M) ⊕ (M ∩ N), ⊥ ⊥ ⊥ R(BAe ) = R(BAe Be) = R(BABe ) ⊕ (M ∩ N ) = (N • M) ⊕ (M ∩ N ), ⊥ ⊥ ⊥ ⊥ ⊥ R(BeAe) = R(BeAeBe) = R(BABe ) ⊕ (M ∩ N ) = (N • M) ⊕ (M ∩ N ).

R( AB + BA ) = R( ABA + BAB ) = R(AB) + R(BA) = (M ◦ N) + (N ◦ M), ⊥ ⊥ R( ABe + BAe ) = R( ABAe + BAe Be ) = R(ABe) + R(BAe ) = (M ◦ N ) + (N ◦ M), ⊥ ⊥ R( ABe + BAe ) = R( ABe Ae + BABe ) = R(ABe ) + R(BAe) = (M ◦ N) + (N ◦ M ), ⊥ ⊥ ⊥ ⊥ R( AeBe + BeAe ) = R( AeBeAe + BeAeBe ) = R(AeBe) + R(BeAe) = (M ◦ N ) + (N ◦ M ), ⊥ ⊥ R( AB + BeAe ) = R( ABA + BeAeBe ) = R(AB) ⊕ R(BeAe) = (M ◦ N) ⊕ (N ◦ M ), ⊥ ⊥ R( ABe + BAe ) = R( ABAe + BABe ) = R(ABe) ⊕ R(BAe) = (M ◦ N ) ⊕ (N ◦ M ), ⊥ ⊥ R( ABe + BAe ) = R( ABe Ae + BAe Be ) = R(ABe ) ⊕ R(BAe ) = (M ◦ N) ⊕ (N ◦ M), ⊥ ⊥ R( AeBe + BA ) = R( AeBeAe + BAB ) = R(AeBe) ⊕ R(BA) = (M ◦ N ) ⊕ (N ◦ M), and

⊥ R( ABAe + ABAe ) = R(ABAe) ⊕ R(ABAe ) = (M • N) ⊕ (M • N), Linear subspaces and orthogonal projectors Ë 171

⊥ R( BABe + BABe ) = R( BABe) ⊕ R(BABe ) = (N • M) ⊕ (N • M), R( ABAe + BABe ) = R(ABAe) ⊕ R(BABe) = (M • N) ⊕ (N • M), ⊥ ⊥ R( ABAe + BABe ) = R(ABAe ) ⊕ R(BABe ) = (M • N) ⊕ (N • M).

(d) The following orthogonal direct sum decomposition identities on the ranges of AB, BA and their operations hold

R( AB − BA ) = R( ABe − BAe ) = R( ABe − BAe ) = R( AeBe − BeAe ) = R( ABA − BAB ) = R( ABAe − BAe Be ) = R( ABe Ae − BABe ) = R( AeBeAe − BeAeBe ) = (M • N) ⊕ (N • M) = (M⊥ • N) ⊕ (N⊥ • M), R( AB + BA ) = R( AB − BA ) ⊕ (M ∩ N) = M • N ⊕ N • M ⊕ (M ∩ N) ⊕ (M ∩ N) = (M⊥ • N) ⊕ (N⊥ • M) ⊕ (M ∩ N), ⊥ ⊥ R( A − B ) = R( AB − BA ) ⊕ (M ∩ N ) ⊕ (M ∩ N) = (M • N) ⊕ (N • M) ⊕ (M ∩ N⊥) ⊕ (M⊥ ∩ N) ⊕ (M ∩ N⊥) = (M⊥ • N) ⊕ (N⊥ • M) ⊕ (M ∩ N⊥) ⊕ (M⊥ ∩ N) ⊕ (M ∩ N⊥), ⊥ ⊥ ⊥ ⊥ R( Im − A − B ) = R( AB + BA ) ⊕ (M ∩ N ) = R( AB − BA ) ⊕ (M ∩ N) ⊕ (M ∩ N ) = (M • N) ⊕ (N • M) ⊕ (M ∩ N) ⊕ (M⊥ ∩ N⊥) = (M⊥ • N) ⊕ (N⊥ • M) ⊕ (M ∩ N) ⊕ (M⊥ ∩ N),

R( Im − AB ) = R( Im − BA ) = R( Im − ABA ) = R( Im − BAB ) ⊥ ⊥ ⊥ ⊥ = R( AB − BA ) ⊕ (M ∩ N ) ⊕ (M ∩ N) ⊕ (M ∩ N ) = (M • N) ⊕ (N • M) ⊕ (M ∩ N⊥) ⊕ (M⊥ ∩ N) ⊕ (M⊥ ∩ N⊥) = (M⊥ • N) ⊕ (N⊥ • M) ⊕ (M ∩ N⊥) ⊕ (M⊥ ∩ N) ⊕ (M⊥ ∩ N⊥), ⊥ ⊥ R( A − BAB ) = R( AB − BA ) ⊕ (M ∩ N ) = (M • N) ⊕ (N • M) ⊕ (M ∩ N ) = (M⊥ • N) ⊕ (N⊥ • M) ⊕ (M ∩ N⊥), ⊥ ⊥ R( B − ABA ) = R( AB − BA ) ⊕ (M ∩ N) = (M • N) ⊕ (N • M) ⊕ (M ∩ N) = (M⊥ • N) ⊕ (N⊥ • M) ⊕ (M⊥ ∩ N).

(e) The following orthogonal direct sum decomposition identities on M, N, and their operations hold

⊥ ⊥ M = R(AB) ⊕ (M ∩ N ) = R(ABe) ⊕ (M ∩ N) = R(ABAe) ⊕ (M ∩ N) ⊕ (M ∩ N ), ⊥ ⊥ N = R(BA) ⊕ (M ∩ N) = R(BAe) ⊕ (M ∩ N) = R(BABe) ⊕ (M ∩ N) ⊕ (M ∩ N), ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ M = R(ABe ) ⊕ (M ∩ N ) = R(AeBe) ⊕ (M ∩ N) = R(ABAe ) ⊕ (M ∩ N) ⊕ (M ∩ N ), ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ N = R(BAe ) ⊕ (M ∩ N ) = R(BeAe) ⊕ (M ∩ N ) = R(BABe ) ⊕ (M ∩ N ) ⊕ (M ∩ N ), ⊥ ⊥ M + N = R( A − B ) ⊕ (M ∩ N) = R( AB + BA ) ⊕ (M ∩ N ) ⊕ (M ∩ N) ⊥ ⊥ = R( AB − BA ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊕ (M ∩ N) ⊥ ⊥ = R(ABAe) ⊕ R( ABAe ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊕ (M ∩ N) ⊥ ⊥ = R(BABe) ⊕ R(BABe ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊕ (M ∩ N) ⊥ ⊥ = R(ABAe) ⊕ R(BABe) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊕ (M ∩ N) ⊥ ⊥ = R(ABAe ) ⊕ R(BABe ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊕ (M ∩ N), ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ M + N = R(ABe + BAe ) ⊕ (M ∩ N ) = [ R(ABe ) + R(BAe )] ⊕ (M ∩ N ) ⊥ ⊥ ⊥ ⊥ = R( A − B ) ⊕ (M ∩ N ) = R( AeBe + BeAe ) ⊕ (M ∩ N ) ⊕ (M ∩ N) 172 Ë Yongge Tian

⊥ ⊥ ⊥ ⊥ = R( AeBe − BeAe ) ⊕ (M ∩ N ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊥ ⊥ ⊥ ⊥ = R(ABAe) ⊕ R( ABAe ) ⊕ (M ∩ N ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊥ ⊥ ⊥ ⊥ = R(BABe) ⊕ R(BABe ) ⊕ (M ∩ N ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊥ ⊥ ⊥ ⊥ = R(ABAe) ⊕ R(BABe) ⊕ (M ∩ N ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊥ ⊥ ⊥ ⊥ = R(ABAe ) ⊕ R(BABe ) ⊕ (M ∩ N ) ⊕ (M ∩ N) ⊕ (M ∩ N ), m ⊥ ⊥ C = R( A − B ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊥ ⊥ ⊥ ⊥ = R( AB + BA ) ⊕ (M ∩ N ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊥ ⊥ = R( AeBe + BeAe ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊕ (M ∩ N) ⊥ ⊥ ⊥ ⊥ = R( AB − BA ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊥ ⊥ ⊥ ⊥ = R(ABAe) ⊕ R( ABAe ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊥ ⊥ ⊥ ⊥ = R(BABe) ⊕ R(BABe ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊥ ⊥ ⊥ ⊥ = R(ABAe) ⊕ R(BABe) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊥ ⊥ ⊥ ⊥ = R(ABAe ) ⊕ R(BABe ) ⊕ (M ∩ N) ⊕ (M ∩ N ) ⊕ (M ∩ N) ⊕ (M ∩ N ),

and the following decomposition identities of orthogonal projectors hold

A = P + P ⊥ = P + P = P + P + P ⊥ , R(AB) M∩N R(AeB) M∩N R(ABeA) M∩N M∩N

B = P + P ⊥ = P + P = P + P + P ⊥ , R(BA) M ∩N R(BeA) M∩N R(BAeB) M∩N M ∩N

A + B = P + P + P ⊥ + P ⊥ = P + P + 2P R(AB) R(BA) M∩N M ∩N R(AeB) R(BeA) M∩N

= P + P + 2P + P ⊥ + P ⊥ , R(ABeA) R(BAeB) M∩N M∩N M ∩N

PM+N = PR( AB+BA ) + PM∩N⊥ + PM⊥∩N = PR( AB−BA ) + PM∩N + PM∩N⊥ + PM⊥∩N

= P + P + P + P ⊥ + P ⊥ R(ABeA) R( eABA) M∩N M∩N M ∩N

= P + P + P + P ⊥ + P ⊥ , R(BAeB) R(eBAB) M∩N M∩N M ∩N

A = P + P ⊥ ⊥ = P + P ⊥ = P + P ⊥ + P ⊥ ⊥ , e R(eAB) M ∩N R(eAeB) M ∩N R(eABA) M ∩N M ∩N

B = P + P ⊥ ⊥ = P + P ⊥ = P + P ⊥ + P ⊥ ⊥ , e R(eBA) M ∩N R(eBeA) M∩N R(eBAB) M∩N M ∩N

A + B = P + P + 2P ⊥ ⊥ = P + P + P ⊥ + P ⊥ e e R(eAB) R(eBA) M ∩N R(eAeB) R(eBeA) M ∩N M∩N

= P + P + P ⊥ + P ⊥ + 2P ⊥ ⊥ , R(eABA) R(eBAB) M ∩N M∩N M ∩N

P ⊥ ⊥ = P + P ⊥ + P ⊥ = P + P ⊥ + P ⊥ + P ⊥ ⊥ M +N R( eAeB+eBeA ) M∩N M ∩N R( eAeB−eBeA ) M∩N M ∩N M ∩N

= P + P + P ⊥ + P ⊥ + P ⊥ ⊥ R(ABeA) R( eABA) M∩N M ∩N M ∩N

= P + P + P ⊥ + P ⊥ + P ⊥ ⊥ , R(BAeB) R(eBAB) M∩N M ∩N M ∩N

Im = PR( A−B ) + PM∩N + PM⊥∩N⊥ = PR( AB+BA ) + PM∩N⊥ + PM⊥∩N + PM⊥∩N⊥

= P + P + P ⊥ + P ⊥ R( eAeB+eBeA ) M∩N M∩N M ∩N

= PR( AB−BA ) + PM∩N + PM∩N⊥ + PM⊥∩N + PM⊥∩N⊥

= P + P + P + P ⊥ + P ⊥ + P ⊥ ⊥ R(ABeA) R( eABA) M∩N M∩N M ∩N M ∩N

= P + P + P + P ⊥ + P ⊥ + P ⊥ ⊥ R(BAeB) R(eBAB) M∩N M∩N M ∩N M ∩N

= P + P + P + P ⊥ + P ⊥ + P ⊥ ⊥ R(ABeA) R(BAeB) M∩N M∩N M ∩N M ∩N

= P + P + P + P ⊥ + P ⊥ + P ⊥ ⊥ . R(eABA) R(eBAB) M∩N M∩N M ∩N M ∩N (f) The following set identities hold

⊥ ⊥ N ( A + B ) = M ∩ N , Linear subspaces and orthogonal projectors Ë 173

⊥ ⊥ N ( A − B ) = (M ∩ N) ⊕ (M ∩ N ), ⊥ ⊥ N ( Im − A − B ) = (M ∩ N ) ⊕ (M ∩ N), ⊥ ⊥ ⊥ ⊥ N ( AB + BA ) = N ( ABA + BAB ) = (M ∩ N ) ⊕ (M ∩ N) ⊕ (M ∩ N ), N ( AB − BA ) = N ( ABA − BAB ) = N ( ABAe + ABAe ) = N ( BABe + BABe ) = N ( ABAe ) ∩ N ( ABAe ) = N ( BABe ) ∩ N (BABe) = (M ∩ N) ⊕ (M ∩ N⊥) ⊕ (M⊥ ∩ N) ⊕ (M⊥ ∩ N⊥),

N ( Im − AB ) = N ( Im − BA ) = N ( Im − ABA ) = N ( Im − BAB ) = M ∩ N, ⊥ ⊥ ⊥ N ( A − BAB ) = (M ∩ N) ⊕ (M ∩ N) ⊕ (M ∩ N ), ⊥ ⊥ ⊥ N ( B − ABA ) = (M ∩ N) ⊕ (M ∩ N ) ⊕ (M ∩ N ).

Based on the given two linear subspaces M and N, we dene eight new linear subspaces from the ranges of the products of A, B, Ae and Be in (3.2) as follows

S1 = R(AB) = R(ABA), T1 = R(BA) = R(BAB), (3.18)

S2 = R(ABe) = R(ABAe ), T2 = R(BAe ) = R(BAe Be), (3.19)

S3 = R(ABe ) = R(ABe Ae), T3 = R(BAe) = R(BABe ), (3.20)

S4 = R(AeBe) = R(AeBeAe), T4 = R(BeAe) = R(BeAeBe). (3.21)

Accordingly, the eight orthogonal projectors on the eight linear subspaces can be represented as

† † A1 = PS1 = (AB)(AB) , B1 = PT1 = (BA)(BA) , (3.22) † † A2 = PS2 = (ABe)(ABe) , B2 = PT2 = (BAe )(BAe ) , (3.23) † † A3 = PS3 = (ABe )(ABe ) , B3 = PT3 = (BAe)(BAe) , (3.24) † † A4 = PS4 = (AeBe)(AeBe) , B4 = PT4 = (BeAe)(BeAe) . (3.25)

Concerning the algebraic properties of these linear subspaces, we have the following results.

m Corollary 3.6. Let M and N be two linear subspaces of C , A, B, Ae and Be in (3.2), Si , Ti , Ai and Bi in (3.18)– (3.25), i = 1, 2, 3, 4. Then, the following results hold.

(a) The products of Ai and Bi , i = 1, 2, 3, 4, satisfy

A1B1 = AB, A1B1A1 = ABA, B1A1B1 = BAB,

A2B2 = ABe, A2B2A2 = ABAe , B2A2B2 = BAe Be,

A3B3 = ABe , A3B3A3 = ABe Ae, B3A3B3 = BABe ,

A4B4 = AeBe, A4B4A4 = AeBeAe, B4A4B4 = BeAeBe.

(b) There exists a unitary matrix U such that " # ! C2 CS A = UdiagI , 0 , I , 0 U∗, B = Udiag , I , 0 U∗, 1 k1 k2 k3 k4+k5+k6 1 CSS2 k3 k4+k5+k6 " # ! S2 −CS A = UdiagI , 0 , I , 0 U∗, B = Udiag , 0 , I , 0 U∗, 2 k1 k2+k3 k4 k5+k6 2 −CSC2 k3 k4 k5+k6 " # ! C2 CS A = Udiag0 , I , 0 , I , 0 U∗, B = Udiag , 0 , I , 0 U∗, 3 k1 k2 k3+k4 k5 k6 3 CSS2 k3+k4 k5 k6 " # ! S2 −CS A = Udiag0 , I , 0 , I U∗, B = Udiag , 0 , I U∗, 4 k1 k2 k3+k4+k5 k6 4 −CSC2 k3+k4+k5 k6 174 Ë Yongge Tian

" # ! I + C2 CS A + B = Udiag k1 , 2I , 0 U∗, 1 1 CSS2 k3 k4+k5+k6 " # ! S2 −CS A − B = Udiag , 0 U∗, 1 1 −CS −S2 m−2k1 " # ! I + S2 −CS A + B = Udiag k1 , 0 , 2I , 0 U∗, 2 2 −CSC2 k3 k4 k5+k6 " # ! C2 CS A − B = Udiag , 0 U∗, 2 2 CS −C2 m−2k1

" 2 # ! C CS ∗ A3 + B3 = Udiag 2 , 0k3+k4 , 2Ik5 , 0k6 U , CSIk2 + S " # ! −C2 −CS A − B = Udiag , 0 U∗, 3 3 −CSC2 m−2k1

" 2 # ! S −CS ∗ A4 + B4 = Udiag 2 , 0k3+k4+k5 , 2Ik6 U , −CSIk2 + C " # ! −S2 CS A − B = Udiag , 0 U∗, 4 4 CSS2 m−2k1 ∗ ∗ A1 − A2 = Udiag 02k1 , Ik3 , −Ik4 , 0k5+k6 U , A3 − A4 = Udiag 02k1+k3+k4 , Ik5 , −Ik6 U , ∗ ∗ B1 − B3 = Udiag 02k1 , Ik3 , 0k4 , −Ik5 , 0k6 U , B2 − B4 = Udiag 02k1+k3 , Ik4 , 0k5 , −Ik6 U , ∗ ∗ A1 + A3 = Udiag I2k1+k3 , 0k4 , Ik5 , 0k6 U , B1 + B2 = Udiag I2k1+k3+k4 , 0k5+k6 U , ∗ ∗ A2 + A4 = Udiag I2k1 , 0k3 , Ik4 , 0k5 , Ik6 U , B3 + B4 = Udiag I2k1 , 0k3+k4 , Ik5+k6 U , ∗ A1 + A4 = B1 + B4 = Udiag I2k1+k3 , 0k4+k5 , Ik6 U , ∗ A2 + A3 = B2 + B3 = Udiag I2k1 , 0k3 , Ik4+k5 , 0k6 U , ∗ A1 + A2 + A3 + A4 = B1 + B2 + B3 + B4 = Im + Udiag I2k1 , 0m−2k1 U .

In particular,

∗ A − A2 = B − B3 = Udiag 02k1 , Ik3 , 0k4+k5+k6 U = PM∩N, ∗ ⊥ A − A1 = Be − B4 = Udiag 02k1+k3 , Ik4 , 0k5+k6 U = PM∩N , ∗ ⊥ Ae − A4 = B − B1 = Udiag 02k1+k3+k4 , Ik5 , 0k6 U = PM ∩N, ∗ ⊥ ⊥ Ae − A3 = Be − B2 = Udiag 0m−k6 , Ik6 U = PM ∩N .

(c) The Moore–Penrose inverses of the matrices in Corollary 3.6(b) are given by " # ! I −CS−1 ( A + B )† = Udiag k1 , 2−1I , 0 U∗, 1 1 −CS−1 S−2 + C2S−2 k3 k4+k5+k6

" −1# ! † Ik1 −CS ∗ ( A1 − B1 ) = Udiag −1 , 0m−2k1 U , −CS −Ik2 " # ! I C−1S ( A + B )† = Udiag k1 , 0 , 2−1I , 0 U∗, 2 2 C−1SC−2 + C−2S2 k3 k4 k5+k6

" −1 # ! † Ik1 C S ∗ ( A2 − B2 ) = Udiag −1 , 0m−2k1 U , C S −Ik2 " −2 −2 2 −1 # ! † C + C S −C S −1 ∗ ( A3 + B3 ) = Udiag −1 , 0k3+k4 , 2 Ik5 , 0k6 U , −CS Ik2 Linear subspaces and orthogonal projectors Ë 175

" −1 # ! † −Ik1 −C S ∗ ( A3 − B3 ) = Udiag −1 , 0m−2k1 U , −C SIk2 " −2 2 −2 −1# ! † S + C S CS −1 ∗ ( A4 + B4 ) = Udiag −1 , 0k3+k4+k5 , 2 Ik6 U , CS Ik2 " −1# ! † −Ik1 CS ∗ ( A4 − B4 ) = Udiag −1 , 0m−2k1 U . CS Ik2

(d) The following dimension identities hold

dim(S1) = dim(T1) = r(A1) = r(B1) = r(AB), dim(S2) = dim(T2) = r(A2) = r(B2) = r(ABe),

dim(S3) = dim(T3) = r(A3) = r(B3) = (ABe ), dim(S4) = dim(T4) = r(A4) = r(B4) = r(AeBe).

(e) The following orthogonal projector identities hold

PS1 = A1, PT1 = B1, PS2 = A2, PT2 = B2, PS3 = A3, PT3 = B3, PS4 = A4, PT4 = B4.

(f) The following intersection identities for subspaces hold

⊥ ⊥ ⊥ ⊥ S1 ∩ T1 = M ∩ N, S2 ∩ T2 = M ∩ N , S3 ∩ T3 = M ∩ N, S4 ∩ T4 = M ∩ N .

Although the results in this section are established in the well-known nite-dimensional complex vector space, most conclusions are obtained only by the usual sum and intersection operations of subspaces. Thus, they can easily be extended to subspaces in various general algebraic structures in which subspaces and their orthogonal complements can be dened.

4 Formulas for orthogonal projectors onto the intersections of two subspaces

Finding explicit expressions of the orthogonal projector onto the intersection of two linear subspaces is a fundamental and interesting problem in linear algebra, which was considered by some authors and many results were obtained, see e.g., in [31, 82, 109, 116, 134, 139]. Although Theorem 3.4(e) give decomposition expressions of the four orthogonal projectors, we cannot clarify explicit contributions of A and B in these representations. In this section, we give a family of known or novel formulas composed by algebraic operations of A and B for representing the orthogonal projectors onto the intersection of two linear subspaces.

m Theorem 4.1. Let M and N be two linear subspaces of C , A, B, Ae, and Be be as given in (3.2), S1 and T1 be as given in (3.18), and Ai and Bi be as given in (3.22)–(3.25), i = 1, ... , 4. Then, the orthogonal projector PM∩N onto the intersection M ∩ N can be written in the following forms

† † PM∩N = A − (ABe)(ABe) = A − A2 = B − (BAe)(BAe) = B − B3 (4.1) 1 = ( A + B − A − B ) (4.2) 2 2 3 † † = A − A(BAe Be) A = B − B(ABe Ae) B (4.3)

† † = ABA − AB(ABe Ae) BA = BAB − BA(BAe Be) AB (4.4) = 2A( A + B )†B = 2(A : B) = 2B( A + B )†A = 2(B : A) (4.5)

= A( A + B )†B + B( A + B )†A (4.6) = 2AB( A + B )†AB = 2BA( A + B )†BA (4.7) 176 Ë Yongge Tian

= AB( A + B )†AB + BA( A + B )†BA (4.8) = 2A − 2A( A + B )†A = 2B − 2B( A + B )†B (4.9)

= A + B − A( A + B )†A − B( A + B )†B (4.10) = 2ABA − 2AB( A + B )†BA = 2BAB − 2BA( A + B )†AB (4.11)

= ABA + BAB − AB( A + B )†BA − BA( A + B )†AB (4.12) = [ A( A + B )†A ]† − A = [ B( A + B )†B ]† − B (4.13) 1 1 1 = [ A( A + B )†A ]† + [ B( A + B )†B ]† − ( A + B ) (4.14) 2 2 2 = A[ B( A + B )†B ]†A − ABA = B[ A( A + B )†A ]†B − BAB (4.15) 1 1 1 = A[ B( A + B )†B ]†A + B[ A( A + B )†A ]†B − ( ABA + BAB ) (4.16) 2 2 2 = ( A + B )( A + B )† − ( A − B )( A − B )† (4.17) † † = [ Im − ( A − B )( A − B ) ]A = [ Im − ( A − B )( A − B ) ]B (4.18) 1 = [ I − ( A − B )( A − B )† ]( A + B ) (4.19) 2 m † † † = ( A + B )( A + B ) − (ABe) − (BAe) (4.20) † † † † = [ Im − (ABe) − (BAe) ]A = [ Im − (ABe) − (BAe) ]B (4.21)

1 † † = [ Im − (ABe) − (BAe) ]( A + B ) (4.22) 2 † † = ( A + B )( A + B ) − A2 − A3 = ( A + B )( A + B ) − B2 − B3 (4.23) 1 = ( A + B )( A + B )† − ( A + A + B + B ) (4.24) 2 2 3 2 3 = A − A( A − B )†A = B − B( B − A )†B (4.25) 1 = [ A + B − A( A − B )†A − B( B − A )†B ] (4.26) 2 = ABA − AB( B − A )†BA = BAB − BA( A − B )†AB (4.27) 1 = [ ABA + BAB( B − A )†BA − BA( A − B )†AB ] (4.28) 2 † † = Im − ( Ae + Be )( Ae + Be ) = Im − ( 2Im − A − B )( 2Im − A − B ) (4.29) † † = Im − ( Im − AB )( Im − AB ) = Im − ( Im − BA )( Im − BA ) (4.30)

† = Im − ( 2Im − AB − BA )( 2Im − AB − BA ) (4.31) = 2A( AB + BA )†A = 2B( AB + BA )†B (4.32)

= A( AB + BA )†A + B( AB + BA )†B (4.33) = 2AB( AB + BA )†BA = 2BA( AB + BA )†AB (4.34)

= AB( AB + BA )†BA + BA( AB + BA )†AB (4.35) = 2(AB)† − 2B( AB + BA )†A = 2(BA)† − 2A( AB + BA )†B (4.36)

= (AB)† + (BA)† − A( AB + BA )†B − B( AB + BA )†A (4.37) = 2AB − 2AB( AB + BA )†AB = 2BA − 2BA( AB + BA )†BA (4.38)

= AB + BA − AB( AB + BA )†AB − BA( AB + BA )†BA (4.39) = ( AB + BA )( AB + BA )† − ( AB − BA )( AB − BA )† (4.40) Linear subspaces and orthogonal projectors Ë 177

† † = Im − ( Im − ABA )( Im − ABA ) = Im − ( Im − BAB )( Im − BAB ) (4.41)

† = Im − ( 2Im − ABA − BAB )( 2Im − ABA − BAB ) (4.42) = 2(ABA)† − 2A( ABA + BAB )†A = 2(BAB)† − 2B( ABA + BAB )†B (4.43)

= (ABA)† + (BAB)† − A( ABA + BAB )†A − B( ABA + BAB )†B (4.44) = 2A(BAB)†A − 2AB( ABA + BAB )†BA = 2B(ABA)†B − 2BA( ABA + BAB )†AB (4.45)

= A(BAB)†A + B(ABA)†B − AB( ABA + BAB )†BA − BA( ABA + BAB )†AB (4.46) = 2A( ABA + BAB )†B = 2B( ABA + BAB )†A (4.47)

= A( ABA + BAB )†B + B( ABA + BAB )†A (4.48) = 2AB( ABA + BAB )†AB = 2BA( ABA + BAB )†BA (4.49)

= AB( ABA + BAB )†AB + BA( ABA + BAB )†BA (4.50) = 2ABA( ABA + BAB )†BAB = 2BAB( ABA + BAB )†ABA (4.51)

= ABA( ABA + BAB )†BAB + BAB( ABA + BAB )†ABA (4.52) = 2ABA − 2ABA( ABA + BAB )†ABA = 2BAB − 2BAB( ABA + BAB )†BAB (4.53)

= ABA + BAB − ABA( ABA + BAB )†ABA − BAB( ABA + BAB )†BAB (4.54) † † † † = (AB)(AB) − (ABAe)(ABAe) = (BA)(BA) − (BABe)( BABe) (4.55)

† † = [ Im − (ABAe)(ABAe) ]AB = [ Im − (BABe)( BABe) ]BA (4.56) = A − A( A − BAB )†A = B − B( B − ABA )†B (4.57) = ABA − AB( B − ABA )†BA = BAB − BA( A − BAB )†AB (4.58) † 1 † † 1 † = AB(AB) − A( BABe + BABe ) A = BA(BA) − B( ABAe + ABAe ) B (4.59) 2 2 1 † 1 † = ABA − AB( ABAe + ABAe ) BA = BAB − BA( BABe + BABe ) AB (4.60) 2 2  †  A 0 Im A 1 1 = ( A + B ) − [ A, B, 0 ] 0 BI  B (4.61) 2 2  m   Im Im 0 0  †  A 0 Im 0     = −2[ 0, 0, Im ] 0 BIm  0  (4.62) Im Im 0 Im  †  ABA 0 Im ABA 1 1 = ( ABA + BAB ) − [ ABA, BAB, 0 ] 0 BABI  BAB (4.63) 2 2  m   Im Im 0 0  †  ABA 0 Im 0     = −2[ 0, 0, Im ] 0 BABIm  0  (4.64) Im Im 0 Im † † = 2A1( A1 + B1 ) B1 = 2(A1 : B1) = 2B1( A1 + B1 ) A1 = 2(B1 : A1) (4.65) † † = A1( A1 + B1 ) B1 + B1( A1 + B1 ) A1 (4.66) † † = 2A1 − 2A1( A1 + B1 ) A1 = 2B1 − 2B1( A1 + B1 ) B1 (4.67) † † = A1 + B1 − A1( A1 + B1 ) A1 − B1( A1 + B1 ) B1 (4.68) 178 Ë Yongge Tian

† † = ( A1 + B1 )( A1 + B1 ) − ( A1 − B1 )( A1 − B1 ) (4.69) † † = [ Im − ( A1 − B1 )( A1 − B1 ) ]A1 = [ Im − ( A1 − B1 )( A1 − B1 ) ]B1 (4.70) 1 = [ I − ( A − B )( A − B )† ]( A + B ) (4.71) 2 m 1 1 1 1 1 1 † † † = ( A1 + B1 )( A1 + B1 ) − (A1Be1) − (B1Ae1) (4.72) † † † = ( A1 + B1 )( A1 + B1 ) − (A1Be1)(A1Be1) − (Ae1B1)(Ae1B1) (4.73) † † † = ( A1 + B1 )( A1 + B1 ) − (B1Ae1)(B1Ae1) − (Be1A1)(Be1A1) (4.74) † † = A1 − (A1Be1)(A1Be1) = B1 − (B1Ae1)(B1Ae1) (4.75)

1 1 † 1 † = ( A1 + B1 ) − (A1Be1)(A1Be1) − (B1Ae1)(B1Ae1) (4.76) 2 2 2 † † = A1 − A1( A1 − B1 ) A1 = B1 − B1( B1 − A1 ) B1 (4.77) 1 1 1 = ( A + B ) − A ( A − B )†A − B ( B − A )†B (4.78) 2 1 1 2 1 1 1 1 2 1 1 1 1 † = Im − ( 2Im − A1 − B1 )( 2Im − A1 − B1 ) (4.79)  †   A 0 Im A 1 1 1 1 = ( A + B ) − [ A , B , 0 ]  0 B I  B  (4.80) 2 1 1 2 1 1  1 m  1 Im Im 0 0  †  A1 0 Im 0     = −2[ 0, 0, Im ] 0 B1 Im  0  (4.81) Im Im 0 Im = lim ( A + B )k /2k (4.82) k→∞ = lim (AB)k = lim (BA)k ([83, 101]) (4.83) k→∞ k→∞ = lim (ABA)k = lim (BAB)k (4.84) k→∞ k→∞ = lim ( AB + BA )k /2k (4.85) k→∞ = lim ( ABA + BAB )k /2k (4.86) k→∞ = A − lim [ A( A + B )†A ]k = B − lim [ B( A + B )†B ]k (4.87) k→∞ k→∞ k k = lim ( A1 + B1 ) /2 (4.88) k→∞ 1 = Im − lim ( 2Im − A − B ) k (4.89) k→∞

1 1 = A − lim (ABAe ) k = B − lim (BABe ) k (4.90) k→∞ k→∞

1 1 = Im − lim ( Im − ABA ) k = Im − lim ( Im − BAB ) k (4.91) k→∞ k→∞ 1 = Im − lim ( 2Im − ABA − BAB ) k . (4.92) k→∞ In consequence, the following statements are equivalent: h1i M ∩ N = {0}. h2i–h41i in Corollary 2.13(a). h42i r[ A, B ] = r(A) + r(B). h43i r(ABe) = r(A) and/or r(BAe) = r(B). h44i R(ABe) = R(A) and/or R(BAe) = R(B). h45i M = S2 and/or N = T3. Linear subspaces and orthogonal projectors Ë 179

h46i A = A2 and/or B = B3. h47i A + B = A2 + B3. h48i A(BAe Be)†A = A and/or B(ABe Ae)†B = B. h49i AB(ABe Ae)†BA = ABA and/or BA(BAe Be)†AB = BAB. h50i A( A + B )†B = 0 and/or B( A + B )†A = 0. h51i A( A + B )†B + B( A + B )†A = 0. h52i AB( A + B )†AB = 0 and/or BA( A + B )†BA = 0. h53i AB( A + B )†AB + BA( A + B )†BA = 0. h54i AB( A + B )†BA = ABA and/or BA( A + B )†AB = BAB. h55i AB( A + B )†BA + BA( A + B )†AB = ABA + BAB. h56i [ A( A + B )†A ]† = A and/or [ B( A + B )†B ]† = B. h57i [ A( A + B )†A ]† + [ B( A + B )†B ]† = A + B. h58i A[ B( A + B )†B ]†A = ABA and/or B[ A( A + B )†A ]† = BAB. h59i A[ B( A + B )†B ]†A + B[ A( A + B )†A ]†B = ABA + BAB. h60i ( A + B )( A + B )† = ( A − B )( A − B )†, i.e., R( A + B ) = R( A − B ). h61i R(A) ⊆ R( A − B ) and/or R(B) ⊆ R( A − B ). h62i ( A + B )( A + B )† = (ABe)† + (BAe)†. h63i [(ABe)† + (BAe)† ]A = A and/or [(ABe)† + (BAe)† ]B = B. h64i [(ABe)† + (BAe)† ]( A + B ) = A + B. † † h65i ( A + B )( A + B ) = A2 + A3 and/or ( A + B )( A + B ) = B2 + B3. † h66i 2( A + B )( A + B ) = A2 + A3 + B2 + B3. h67i A( A − B )†A = A and/or B( B − A )†B = B. h68i A( A − B )†A + B( B − A )†B = A + B. h69i AB( B − A )†BA = ABA and/or BA( A − B )†AB = BAB. h70i AB( B − A )†BA + BA( A − B )†AB = ABA + BAB. h71i A + B ≺ 2Im . h72i r( Im − AB ) = r( Im − BA ) = m. h73i AB + BA ≺ 2Im . h74i A( AB + BA )†A = 0 and/or B( AB + BA )†B = 0. h75i A( AB + BA )†A + B( AB + BA )†B = 0. h76i AB( AB + BA )†BA = 0 and/or BA( AB + BA )†AB = 0. h77i AB( AB + BA )†BA + BA( AB + BA )†AB = 0. h78i (AB)† = 2B( AB + BA )†A and/or (BA)† = 2A( AB + BA )†B. h79i (AB)† + (BA)† = A( AB + BA )†B − B( AB + BA )†A. h80i AB( AB + BA )†AB = AB and/or BA( AB + BA )†BA = BA. h81i AB( AB + BA )†AB − BA( AB + BA )†BA = AB + BA. h82i R( AB + BA ) = R( AB − BA ). h83i ABA ≺ Im and/or BAB ≺ Im . h84i ABA + BAB ≺ 2Im . h85i (ABA)† = A( ABA + BAB )†A and/or (BAB)† = B( ABA + BAB )†B. h86i (ABA)† + (BAB)† = A( ABA + BAB )†A + B( ABA + BAB )†B. h87i A(BAB)†A = AB( ABA + BAB )†BA and/or B(ABA)†B = BA( ABA + BAB )†AB. h88i A(BAB)†A + B(ABA)†B = AB( ABA + BAB )†BA + BA( ABA + BAB )†AB. h89i A( ABA + BAB )†B = 0 and/or B( ABA + BAB )†A = 0. h90i A( ABA + BAB )†B + B( ABA + BAB )†A = 0. h91i AB( ABA + BAB )†AB = 0 and/or BA( ABA + BAB )†BA = 0. h92i AB( ABA + BAB )†AB + BA( ABA + BAB )†BA = 0. h93i ABA( ABA + BAB )†BAB = 0 and/or BAB( ABA + BAB )†ABA = 0. h94i ABA( ABA + BAB )†BAB + BAB( ABA + BAB )†ABA = 0. h95i ABA( ABA + BAB )†ABA = ABA and/or BAB( ABA + BAB )†BAB = BAB. h96i ABA( ABA + BAB )†ABA + BAB( ABA + BAB )†BAB = ABA + BAB. 180 Ë Yongge Tian h97i (ABAe)(ABAe)† = (AB)(AB)† and/or (BABe)( BABe)† = (BA)(BA)†. h98i A( A − BAB )†A = A and/or B( B − ABA )†B = B. h99i AB( B − ABA )†BA = ABA and/or BA( A − BAB )†AB = BAB. h100i R(AB) = R(ABAe) and/or R(BA) = R(BABe). h101i A( A − BAB )†A = A and/or B( B − ABA )†B = A. h102i A( A − BAB )†A + B( B − ABA )†B = A + B. h103i AB( B − ABA )†BA = ABA and/or BA( A − BAB )†AB = BAB. h104i AB( B − ABA )†BA + BA( A − BAB )†AB = ABA + BAB. h105i A( BABe + BABe )†A = 2AB(AB)† and/or B( ABAe + ABAe )†B = 2BA(BA)†. h106i AB( ABAe + ABAe )†BA = 2ABA and/or BA( BABe + BABe )†AB = 2BAB.  †  A 0 Im A     h107i [ A, B, 0 ] 0 BIm B = A + B. Im Im 0 0  †  A 0 Im 0     h108i [ 0, 0, Im ] 0 BIm  0  = 0. Im Im 0 Im  †  ABA 0 Im ABA     h109i [ ABA, BAB, 0 ] 0 BABIm BAB = ABA + BAB. Im Im 0 0  †  ABA 0 Im 0     h110i [ 0, 0, Im ] 0 BABIm  0  = 0. Im Im 0 Im h111i S1 ∩ T1 = {0}, i.e., R(AB) ∩ R(BA) = {0}. h112i r[ A1, B1 ] = r(A1) + r(B1). i.e., r[ AB, BA ] = 2r(AB). † † h113i A1( A1 + B1 ) B1 = 0 and/or B1( A1 + B1 ) A1 = 0. † † h114i A1( A1 + B1 ) B1 + B1( A1 + B1 ) A1 = 0. † † h115i A1( A1 + B1 ) A1 = A1 and/or B1( A1 + B1 ) B1 = B1. † † h116i A1( A1 + B1 ) A1 + B1( A1 + B1 ) B1 = A1 + B1. h117i R( A1 + B1 ) = R( A1 − B1 ). † † † h118i ( A1 + B1 )( A1 + B1 ) = (A1Be1) + (B1Ae1) . † † † † † h119i ( A1 + B1 )( A1 + B1 ) = (A1Be1)(A1Be1) + (Ae1B1)(Ae1B1) and/or ( A1 + B1 )( A1 + B1 ) = (B1Ae1)(B1Ae1) + † (Be1A1)(Be1A1) . † † h120i (A1Be1)(A1Be1) = A1 and/or (B1Ae1)(B1Ae1) = B1. h121i R(A1Be1) = R(A1) and/or R(B1Ae1) = R(B1). † † h122i (A1Be1)(A1Be1) + (B1Ae1)(B1Ae1) = A1 + B1. † † h123i A1( A1 − B1 ) A1 = A1 and/or B1( B1 − A1 ) B1 = B1. † † h124i A1( A1 − B1 ) A1 + B1( B1 − A1 ) B1 = A1 + B1. h125i A1 + B1 ≺ 2Im .  †  A1 0 Im A1     h126i [ A1, B1, 0 ] 0 B1 Im B1 = A1 + B1. Im Im 0 0  †  A1 0 Im 0     h127i [ 0, 0, Im ] 0 B1 Im  0  = 0. Im Im 0 Im k k h128i limk→∞( A + B ) /2 = 0. k k h129i limk→∞(AB) = 0 and/or limk→∞(BA) = 0. k k h130i limk→∞(ABA) = 0 and/or limk→∞(BAB) = 0. k k h131i limk→∞( AB + BA ) /2 = 0. k k h132i limk→∞( ABA + BAB ) /2 = 0. Linear subspaces and orthogonal projectors Ë 181

† k † k h133i limk→∞[ A( A + B ) A ] = A and/or limk→∞[ B( A + B ) B ] = B. k k h134i limk→∞( A1 + B1 ) /2 = 0. 1 h135i limk→∞( 2Im − A − B ) k = Im . 1 1 h136i limk→∞(ABAe ) k = A and/or limk→∞(BABe ) k = B. 1 1 h137i limk→∞( Im − ABA ) k = Im and/or limk→∞( Im − BAB ) k = Im . 1 h138i limk→∞( 2Im − ABA − BAB ) k = Im . h139i There exists an A(1) such that A(1)B = 0 and/or there exists a B(1) such that B−A = 0.

Proof. Eqs. (4.1) and (4.2) follow from A − A2 = B − B3 in Corollary 3.6(b). The following identities can be veried from (3.3)–(3.7), Theorem 3.4(a), and the denition of the Moore–Penrose inverse † † −1 ∗ A( A + B ) B = B( A + B ) A = Udiag 02k1 , 2 Ik3 , 0k4+k5+k6 U , (4.93)

† −1 ∗ A( A + B ) A = Udiag Ik1 , 0k2 , 2 Ik3 , Ik4 , 0k5+k6 U , (4.94) " # ! C2 CS B( A + B )†B = Udiag , 2−1I , 0, I , 0 U∗, (4.95) CSS2 k3 k5 k6 " # ! I + C2 CS A( A + B )†A + B( A + B )†B = Udiag k1 , I , 0 U∗, (4.96) CSS2 k3+k4+k5 k6 † † ∗ [ A( A + B ) A ] = Udiag Ik1 , 0k2 , 2Ik3 , Ik4 , 0k5+k6 U , (4.97) " # ! C2 CS [ B( A + B )†B ]† = Udiag , 2I , 0, I , 0 U∗, (4.98) CSS2 k3 k5 k6 † ∗ A( A − B ) A = Udiag Ik1 , 0k2+k3 , Ik4 , 0k5+k6 U , (4.99) " # ! C2 CS B( A − B )†B = −Udiag , 0 , I , 0 U∗, (4.100) CSS2 k3+k4 k5 k6

† † −1 ∗ A( AB + BA ) A = BA( AB + BA ) AB = Udiag 02k1 , 2 Ik3 , 0k4+k5+k6 U , (4.101)

† † −1 ∗ B( AB + BA ) B = AB( AB + BA ) BA = Udiag 02k1 , 2 Ik3 , 0k4+k5+k6 U , (4.102) † † † † A(BAB) A = AB(AB) = A1, B(ABA) B = BA(BA) = B1, (4.103) † † † † A(BAe Be) A = (ABe)(ABe) = A2, Be(ABAe ) Be = (BAe )(BAe ) = B2, (4.104) † † † † Ae(BABe ) Ae = ABe (ABe ) = A3, B(ABe Ae) B = (BAe)(BAe) = B3, (4.105) † † † † Ae(BeAeBe) Ae = (AeBe)(AeBe) = A4, Be(AeBeAe) Be = (BeAe)(BeAe) = B4, (4.106) † −2 −1 ∗ A( ABA + BAB ) A = Udiag C , 0k2 , 2 Ik3 , 0k4+k5+k6 U , (4.107) " # ! I C−1S B( ABA + BAB )†B = Udiag k1 , 2−1I , 0 U∗, (4.108) C−1SC−2S2 k3 k4+k5+k6

† −1 ∗ A( ABA + BAB ) B = Udiag 0k1+k2 , 2 Ik3 , 0k4+k5+k6 U , (4.109) † ∗ A( A − BAB ) A = Udiag Ik1 , 0k2+k3 , Ik4 , 0k5+k6 U , (4.110) " # ! C2 CS B( B − ABA )†B = Udiag , 0 , I , 0 U∗. (4.111) CSS2 k3+k4 k5 k6

Substituting (4.93)–(4.111) into (4.3)–(4.60), and compare the results obtained with the rst equality in Theo- rem 3.4(e), we obtain the identities in (4.3)–(4.60). m m×n In order to show (4.61) and (4.62), we need a result in [137]: for 0 4 A ∈ CH and B ∈ C , the following expansion formula holds † " # " † † ∗ † † ∗ # AB (EB AEB) (B ) − (EB AEB) A(B ) ∗ = † † † † † ∗ † † † ∗ . (4.112) B 0 B − B A(EB AEB) −B A(B ) + B A(EB AEB) A(B ) 182 Ë Yongge Tian

" # " # A 0 I Let D = and H = m . Then we obtain from (4.112) that 0 B Im

 † A 0 Im " # " # S ∗ † 1 † 1 Im −Im  0 BIm = , H = [ Im , Im ], EH = I2m − HH = , (4.113)   ∗ T 2 2 −Im Im Im Im 0 where

† " 1 1 # " † †# † 4 ( A + B ) − 4 ( A + B ) ( A + B ) −( A + B ) S = (EH DEH) = 1 1 = † † , − 4 ( A + B ) 4 ( A + B ) −( A + B ) ( A + B ) † † ∗ † † † ∗ T = −H D(H ) + H D(EH DEH) D(H ) 1 1 1 1 1 = − ( A + B ) + A( A + B )†A + B( A + B )†B − A( A + B )†B − B( A + B )†A, 4 4 4 4 4 so that  †  A 0 Im A 1 1 1 A + B − [ A, B, 0 ] 0 BI  B 2 2 2  m   Im Im 0 0 " # 1 1 1 A = A + B − [ A, B ]S 2 2 2 B 1 1 1 1 1 1 = A + B − A( A + B )†A − B( A + B )†B + A( A + B )†B + B( A + B )†A 2 2 2 2 2 2

= PM∩N ((by (4.1) and (4.6)–(4.10)), and  †  A 0 Im 0 1 1 1 − 2[ 0, 0, I ] 0 BI   0  = ( A + B ) − A( A + B )†A − B( A + B )†B m  m   2 2 2 Im Im 0 Im 1 1 + A( A + B )†B + B( A + B )†A 2 2

= PM∩N ((by (4.1) and (4.6)–(4.10)), establishing the identities of (4.1), (4.61) and (4.62). Similarly, we obtain from (4.113), (4.1), and (4.49)–(4.52) that

 †  ABA 0 Im ABA 1 1 1 ABA + BAB − [ ABA, BAB, 0 ] 0 BABI  BAB 2 2 2  m   Im Im 0 0 1 1 1 1 = ABA + BAB − ABA( ABA + BAB )†ABA − BAB( ABA + BAB )†BAB 2 2 2 2 1 1 + ABA( ABA + BAB )†BAB + BAB( ABA + BAB )†ABA 2 2

= PM∩N (by (4.1) and (4.49)–(4.52))  †  ABA 0 Im 0     = − 2[ 0, 0, Im ] 0 BABIm  0  Im Im 0 Im 1 1 1 1 = ABA + BAB − ABA( ABA + BAB )†ABA − BAB( ABA + BAB )†BAB 2 2 2 2 1 1 + ABA( ABA + BAB )†BAB + BAB( ABA + ABA )†ABA 2 2 Linear subspaces and orthogonal projectors Ë 183

= PM∩N (by (4.1) and (4.49)–(4.52)), establishing the identities of (4.1), (4.63) and (4.64). Replacing the two orthogonal projectors A and B with the two orthogonal projectors A1 and B1, and observing from Corollary 3.6(f) that PM∩N = PS1∩T1 , we obtain (4.65)–(4.81) from (4.1)–(4.62). Matrix decomposition identities of ( A + B )k, (AB)k, (ABA)k, (BAB)k, ( AB + BA )k, ( ABA + BAB )k, [ A( A + † k † k k 1/k 1/k 1/k 1/k 1/k B ) A ] , [ B( A+B ) B ] , ( A1 +B1 ) , (ABAe ) , (BABe ) , (Im −ABA) , (Im −BAB) and (Im −ABA−BAB) can be established from Theorem 3.4(a), (4.94), and (4.95). Taking the limits of these matrix decomposition identities yields the equivalence of (4.1), (4.82)–(4.92). Note that dim((M + N) = r[ A, B ], dim(M) = r(A) and dim(N) = r(B). Hence Statements h27i and h35i are equivalent. Applying the following well-known results in [103]

r[ X, Y ] = r(X) + r( Y − XX†Y ) = r(Y) + r( X − YY†X ), and

r[ X, Y ] = r(X) + r(Y) ⇔ r( Y − XX†Y ) = r(Y) ⇔ r( X − YY†X ) = r(X) ∗ ∗ † ∗ ∗ ∗ † ∗ ⇔ R( Y − Y XX ) = R(Y ) ⇔ r( X − X YY ) = r(X ) to [ A, B ] lead to the equivalence of h42i–h44i. Setting (4.1)–(4.92) equal to zero yields the equivalences of h1i, and h45i–h138i. The equivalence of h42i and h139i follows from the following known formulas

min r(A(1)B) = min r(B(1)A) = r(A) + r(B) − r[ A, B ], A(1) B(1) see [133].

Some of the expressions in (4.1)–(4.92) were given in the literature; see e.g., [31, 49, 109, 116, 136]. Replacing M and N with M⊥ and N⊥ respectively in Theorem 4.1 will yield other three families of formulas for representing the orthogonal projectors onto the subspace intersections M ∩ N⊥, M⊥ ∩ N, and M⊥ ∩ N⊥, respectively, as well as, a huge amount (1394) of identifying conditions for M and N to be in generic position. Here we select a group of such conditions as the representatives.

m Theorem 4.2. Let M and N be two linear subspaces of C with m even, A = PM, B = PN, Si , and Ti be as given in (3.18)–(3.21), i = 1, 2, 3, 4. Then, the following statements are equivalent: h1i M and N are in generic position. ⊥ ⊥ ⊥ ⊥ m h2i M + N = M + N = M + N = M + N = C . h3i M = M • N, N = N • M, M⊥ = M⊥ • N and N⊥ = N⊥ • M. ⊥ ⊥ ⊥ ⊥ m h4i (M • N) ⊕ (N • M) = (M • N) ⊕ (N • M) = (M • N) ⊕ (N • M) = (M • N) ⊕ (N • M) = C . h5i Si ∩ Ti = {0}, i = 1, 2, 3, 4. h6i r[ A, B ] = 2r(A) = 2r(B) = 2r(AB) = m. h7i A + B, Ae − B, and Ae + Be are nonsingular. h8i A + B, Ae − B, and 2Im − AB − BA are nonsingular. h9i A + B, Ae − B, and 2Im − ABA − BAB are nonsingular. h10i Both Ae − B is nonsingular and −Im ≺ Ae − B ≺ Im . 2 h11i 0 ≺ ( Ae − B ) ≺ Im . h12i −Im ≺ A − B ≺ Im and A − B is nonsingular. 2 h13i 0 ≺ ( A − B ) ≺ Im . h14i AB − BA is nonsingular. h15i ABA − BAB is nonsingular. h16i There exist a unitary matrix U and a diagonal matrix C with 0 ≺ C ≺ Im/2 such that

" # " 2 2 1/2# Im/2 0 ∗ C C(Im/2 − C ) ∗ A = U U and B = U 2 1/2 2 U . 0 0 C(Im/2 − C ) Im/2 − C 184 Ë Yongge Tian

h17i There exist a unitary matrix U and a diagonal matrix C with 0 ≺ C ≺ Im/2 such that ∗ A + B = Udiag Im/2 − C, Im/2 + C U . h18i There exist a unitary matrix U and a diagonal matrix C with 0 ≺ S ≺ Im/2 such that

A − B = Udiag (−S, S) U∗. h19i There exist a unitary matrix U and a diagonal matrix C with 0 ≺ C ≺ Im/2 such that " # C2 C(I − C2)1/2 AB = U m/2 U∗. 0 0 h20i There exist a unitary matrix U and a diagonal matrix C with 0 ≺ C ≺ Im/2 such that ∗ AB + BA = Udiag −C( Im/2 − C ), C( Im/2 + C ) U . h21i There exist a unitary matrix U and a diagonal matrix C with 0 ≺ C ≺ Im/2 such that

" 2 1/2# 0 C(Im/2 − C ) ∗ AB − BA = U 2 1/2 U . −C(Im/2 − C ) 0 h22i There exist a unitary matrix U and a diagonal matrix C with 0 ≺ C ≺ Im/2 such that

2 2 2 2 2 ∗ ( AB − BA ) = Udiag −C (Im/2 − C ), −C (Im/2 − C ) U . h23i There exist a unitary matrix U and a diagonal matrix C with 0 ≺ C ≺ Im/2 such that ABA = U C2, 0 U∗. h24i There exist a unitary matrix U and a diagonal matrix C with 0 ≺ S ≺ Im/2 such that A − ABA = Udiag S2, 0 U∗. h25i There exist a unitary matrix U and a diagonal matrix C with 0 ≺ C ≺ Im/2 such that

2 2 ∗ ABA + BAB = Udiag C ( Im/2 − C ), C ( Im/2 + C ) U . h26i There exist a unitary matrix U and a diagonal matrix C with 0 ≺ C ≺ Im/2 such that

2 2 1/2 2 2 1/2 ∗ ABA − BAB = Udiag −C ( Im/2 − C ) , C ( Im/2 − C ) U .

It is obvious that the parts of the matrices on the left-hand sides of the equalities in Theorem 3.4(a) are special cases of the following linear combination

M = a0Im + a1A + a2B + a3AB + a4BA + a5ABA + a6BAB, (4.114) where a0, a1, ... , a6 ∈ C. The special cases aPM+bPN, aPM+bPN+cPMPN, as well as, aPM+bPN+cPM∩N were widely approached, in particular, people are interested in decomposing a matrix as given in (4.114), see e.g., [28, 29, 42, 65, 86, 92, 111, 125, 129, 132, 138, 141–143, 154]. By (3.3) and (3.4), the matrix M in (4.114) can be decomposed as ∗ M = Udiag N, aIk3 , ( a0 + a1 )Ik4 , ( a0 + a2 )Ik5 , a0Ik6 U , where a = a0 + a1 + a2 + a3 + a4 + a5 + a6, and

" 2 4 3 # ( a0 + a1 )Ik1 + ( a2 + a3 + a4 + a5 )C + a6C ( a2 + a3 )CS + a6C S N = 3 2 2 2 . ( a2 + a4 )CS + a6C S a0Ik2 + a2S + a6C S Hence, this fact shows that all behaviors of the M can be characterized from its decomposition. We next give some consequences as illustration. Linear subspaces and orthogonal projectors Ë 185

(a) The determinant of M is

k6 k4 k5 k3 |M| = a0 ( a0 + a1 ) ( a0 + a2 ) ( a0 + a1 + a2 + a3 + a4 + a5 + a6 ) |N|, where h i h i |N| = ( a + a )I + ( a + a + a + a )C2 + a C4 a I + a S2 + a C2S2 0 1 k1 2 3 4 5 6 0 k2 2 6 h i h i 3 3 − ( a2 + a3 )CS + a6C S ] ( a2 + a4 )CS + a6C S . (b) The rank of M is

r(M) = r(N) + k3r(a) + k4r( a0 + a1 ) + k5r( a0 + a2 ) + k6r(a0).

† n † † † † † o ∗ M = Udiag N , a Ik3 , ( a0 + a1 ) Ik4 , ( a0 + a2 ) Ik5 , a0Ik6 U .

In particular, M is nonsingular if and only if a0( a0 + a1 )( a0 + a2 )a ≠ 0 and |N| ≠ 0. In this case,

−1 n −1 −1 −1 −1 −1 o ∗ M = Udiag N , a Ik3 , ( a0 + a1 ) Ik4 , ( a0 + a2 ) Ik5 , a0 Ik6 U .

(d) Formulas for calculating the eigenvalues of N can be derived. In particular, under the conditions a0, a1, a2, a5, a6 ∈ R and a3 ∈ C, the linear combination

M = a0Im + a1A + a2B + a3AB + a3BA + a5ABA + a6BAB

is Hermitian, and the partial inertia of M is

i±(M)

= i±(N) + k3i±( a0 + a1 + a2 + a3 + a3 + a5 + a6 ) + k4i±( a0 + a1 ) + k5i±( a0 + a2 ) + k6i±(a0),

where

" 2 4 3 # ( a0 + a1 )Ik1 + ( a2 + a3 + a3 + a5 )C + a6C ( a2 + a3 )CS + a6C S N = 3 2 2 2 . ( a2 + a3 )CS + a6C S a0Ik2 + a2S + a6C S Formulas for calculating the real eigenvalues of the matrix N can be established. In addition, more general linear combinations of the two orthogonal projectors A, B, and their products can be formulated, such as,

2 2 a0Im + a1A + a2B + a3AB + a4BA + a5ABA + a6BAB + a7(AB) + a8(BA) + ··· , see e.g., [44–46, 132]. It is not dicult to describe properties of such kind of matrix expressions and to establish fruitful results and facts on two orthogonal projectors and their algebraic operations.

5 More on commutativity of two subspaces and two orthogonal projectors

Commutativity of two linear subspaces, or equivalently, the commutative identity AB = BA for the orthogonal projectors A = PM and B = PN is a quite special topic that attracts much attention in matrix theory, which has essential connections with many facts in matrix analysis and important applications in mathematics. There are many results on commutativity of two orthogonal projectors in the literature; see, e.g., [5, 7, 8, 10– 12, 14, 15, 18, 19, 21, 33–35, 76, 85, 93, 112, 122, 123, 132, 135, 140, 147] and the relevant literature quoted there. In particular, it has closed link with reverse-order law problem (MN)(i,...,j) = N(k,...,l)M(s,...,t) for generalized inverses of matrix product MN. To illustrate this link, we present a known fact. 186 Ë Yongge Tian

m×n n×p Lemma 5.1 ([89,140]). Let M ∈ C and N ∈ C . Then the following statements are equivalent:

h1i {(MN)(1)} ⊇ {N†M(1)}. h2i {(MN)(1)} ⊇ {N†M(1,2)}. h3i {(MN)(1)} ⊇ {N†M(1,3)}. h4i {(MN)(1)} ⊇ {N†M(1,4)}. h5i {(MN)(1)} ⊇ {N†M(1,2,3)}. h6i {(MN)(1)} ⊇ {N†M(1,2,4)}. h7i {(MN)(1)} ⊇ {N†M(1,3,4)}. h8i {(MN)(1)} ⊇ {N(1,3,4)M(1)}. h9i {(MN)(1)} ⊇ {N(1,3,4)M(1,2)}. h10i {(MN)(1)} ⊇ {N(1,3,4)M(1,3)}. h11i {(MN)(1)} ⊇ {N(1,3,4)M(1,4)}. h12i {(MN)(1)} ⊇ {N(1,3,4)M(1,2,3)}. h13i {(MN)(1)} ⊇ {N(1,3,4)M(1,2,4)}. h14i {(MN)(1)} ⊇ {N(1,3,4)M(1,3,4)}. h15i {(MN)(1)} ⊇ {N(1,3,4)M†}. h16i {(MN)(1)} ⊇ {N(1,2,4)M(1,4)} h17i {(MN)(1)} ⊇ {N(1,2,4)M(1,2,4)}. h18i {(MN)(1)} ⊇ {N(1,2,4)M(1,3,4)}. h19i {(MN)(1)} ⊇ {N(1,2,4)M†}. h20i {(MN)(1)} ⊇ {N(1,2,3)M(1)}. h21i {(MN)(1)} ⊇ {N(1,2,3)M(1,2)}. h22i {(MN)(1)} ⊇ {N(1,2,3)M(1,3)}. h23i {(MN)(1)} ⊇ {N(1,2,3)M(1,4)}. h24i {(MN)(1)} ⊇ {N(1,2,3)M(1,2,3)}. h25i {(MN)(1)} ⊇ {N(1,2,3)M(1,2,4)}. h26i {(MN)(1)} ⊇ {N(1,2,3)M(1,3,4)}. h27i {(MN)(1)} ⊇ {N(1,2,3)M†}. h28i {(MN)(1)} ⊇ {N(1,4)M(1,4)}. h29i {(MN)(1)} ⊇ {N(1,4)M(1,2,4)}. h30i {(MN)(1)} ⊇ {N(1,4)M(1,3,4)}. h31i {(MN)(1)} ⊇ {N(1,4)M†}. h32i {(MN)(1)} ⊇ {N(1,3)M(1)}. h33i {(MN)(1)} ⊇ {N(1,3)M(1,2)}. h34i {(MN)(1)} ⊇ {N(1,3)M(1,3)}. h35i {(MN)(1)} ⊇ {N(1,3)M(1,4)}. h36i {(MN)(1)} ⊇ {N(1,3)M(1,2,3)}. h37i {(MN)(1)} ⊇ {N(1,3)M(1,2,4)}. h38i {(MN)(1)} ⊇ {N(1,3)M(1,3,4)}. h39i {(MN)(1)} ⊇ {N(1,3)M†}. h40i {(MN)(1)} ⊇ {N(1,2)M(1,4)}. h41i {(MN)(1)} ⊇ {N(1,2)M(1,2,4)}. h42i {(MN)(1)} ⊇ {N(1,2)M(1,3,4)}. h43i {(MN)(1)} ⊇ {N(1,2)M†}. h44i {(MN)(1)} ⊇ {N(1)M(1,4)}. h45i {(MN)(1)} ⊇ {N(1)M(1,2,4)}. h46i {(MN)(1)} ⊇ {N(1)M(1,3,4)}. h47i {(MN)(1)} ⊇ {N(1)M†}. h48i {(MN)(1)} 3 N†M†. h49i MNN†M†MN = MN. h50i (M†M)(NN†) = (NN†)(M†M).

Lemma 5.1 shows that the matrix set inclusions and matrix equality problem in h1i–h49i are equivalent to the commutativity of the two orthogonal projectors M†M and NN†. Based on the results in the previous sections, we obtain by observation the following equivalent statements for the commutative identity AB = BA to hold, some of which were established in the references mentioned above.

m Theorem 5.2. Let M and N be two linear subspaces in C , A, B, Ae, and Be be as given in (3.2), s be an integer, and k3, k4, k5, k6 be as given in (3.8) and (3.9), respectively. Then, the following statements are equivalent: h1i M and N commute. h2i–h24i in Corollary 2.14. h25i Some/all of the four facts: AB = BA, ABe = BAe , ABe = BAe, AeBe = BeAe. h26i Some/all of the four facts: ABA = BAB, ABAe = BAe Be, ABe Ae = BABe , AeBeAe = BeAeBe. h27i There exists a unitary matrix U such that

∗ ∗ A = Udiag Ik3+k4 , 0k5+k6 U and B = Udiag Ik3 , 0k4 , Ik5 , 0k6 U . h28i There exists a unitary matrix U such that

∗ αA + βB = Udiag (α + β)Ik3 , αIk4 , βIk5 , 0k6 U

for some/all two nonzero real numbers α and β. h29i There exists a unitary matrix U such that

∗ ∗ AB = Udiag Ik3 , 0m−k3 U and/or BA = Udiag Ik3 , 0m−k3 U ,

namely, AB and/or BA is an orthogonal projector. Linear subspaces and orthogonal projectors Ë 187 h30i There exists a unitary matrix U such that

∗ αAB + βBA = Udiag (α + β)Ik3 , 0m−k3 U

for some/all two nonzero real numbers α and β. h31i There exists a unitary matrix U such that

∗ ∗ ABA = Udiag Ik3 , 0m−k3 U and/or BAB = Udiag Ik3 , 0m−k3 U ,

namely, ABA and/or BAB is an orthogonal projector. h32i There exists a unitary matrix U such that

∗ αABA + βBAB = Udiag (α + β)Ik3 , 0m−k3 U

for some/all two nonzero real numbers α and β. h33i There exists a unitary matrix U such that

∗ αA + βBAB = Udiag (α + β)Ik3 , αIk4 , 0k5+k6 U

and/or ∗ αB + βABA = Udiag (α + β)Ik3 , 0k4 , αIk5 , 0k6 U for some/all two nonzero real numbers α and β. h34i There exists a unitary matrix U such that

† † ∗ α(AB) + β(BA) = Udiag (α + β)Ik3 , 0m−k3 U

for some/all two nonzero real numbers α and β. h35i There exists a unitary matrix U such that

† † ∗ α(ABA) + β(BAB) = Udiag (α + β)Ik3 , 0m−k3 U

for some/all two nonzero real numbers α and β. 2 2 h36i ( A + B ) < ( A − B ) , i.e., AB + BA < 0. 2 2 h37i ( A − B ) < A − B and/or ( B − A ) < B − A. 2 m h38i ( A − B ) ∈ COP. 2s+1 h39i ( A − B ) = A − B for some/all s > 1. h40i ( A − B )† = A − B. h41i A( A − B )† = ( A − B )†A and/or B( A − B )† = ( A − B )†B. h42i A( A − B )† = ( A − B )†A and/or B( A − B )† = ( A − B )†B. h43i ( A + B )( A − B )† = ( A − B )†( A + B ). h44i A( A + B )†A + B( A + B )†B is an orthogonal projector. h45i A( A + B )† = ( A + B )†A and/or B( A + B )† = ( A + B )†B. 2 2 h46i ( Im − A − B ) < Im − A − B and/or ( A + B − Im ) < A + B − Im . h47i ( I − A − B )2 is an orthogonal projector. 2s+1 h48i ( Im − A − B ) = Im − A − B for some/all s > 1. † h49i ( Im − A − B ) = Im − A − B. † † h50i ( A − B )( Im − A − B ) = ( Im − A − B ) ( A − B ). h51i ( A + B )( A + B )† = A + B − AB and/or ( A + B )( A + B )† = A + B − BA. h52i [( A + B )( A + B )† − A ][ ( A + B )( A + B )† − B ] = 0 and/or [( A + B )( A + B )† − B ][ ( A + B )( A + B )† − A ] = 0. h53i ( A + B )( A + B )† = A + ABe Ae and/or ( A + B )( A + B )† = B + BAe Be. h54i Some/all of the four facts: A + B = ( A − B )( A − B )† + 2(AB)(AB)†, A + B = ( A − B )( A − B )† + 2(BA)(BA)†, A + B = ( A − B )( A − B )† + 2(AB)†(AB), A + B = ( A − B )( A − B )† + 2(BA)†(BA). † † † † h55i Some/all of the four facts: A < (BA)(BA) , A < (AB) (AB), B < (AB)(AB) , B < (BA) (BA). † † † † h56i A + B < (AB)(AB) + (BA)(BA) and/or A + B < (AB) (AB) + (BA) (BA). 188 Ë Yongge Tian

† † † † h57i Some/all of the four facts: A < (BAe )(BAe ) , A < (ABe) (ABe), B < (ABe )(ABe ) , B < (BAe) (BAe). † † † † h58i A + B < (ABe )(ABe ) + (BAe )(BAe ) and/or A + B < (BAe) (BAe) + (ABe) (ABe). h59i 2A( AB + BA )†B = AB and/or 2B( AB + BA )†A = BA. h60i 2A( AB + BA )†B + 2B( AB + BA )†A = AB + BA. s m h61i (AB) ∈ COP for some/all s > 2. s h62i (AB) = AB for some/all s > 2. s s h63i (AB) = (BA) for some/all s > 1. s h64i (AB) = BA for some/all s > 2. s s h65i (AB) + (BA) = AB + BA for some/all s > 2. s m s m h66i (ABA) ∈ COP and/or (BAB) ∈ COP for some/all s > 2. s s h67i (ABA) = ABA and/or (BAB) = BAB for some/all s > 2. s s h68i (ABA) = BAB and/or (BAB) = ABA for some/all s > 2. s s h69i (ABA) + (BAB) = ABA + BAB for some/all s > 2. s s h70i (ABA) = AB and/or (BAB) = BA for some/all s > 1. s s h71i (ABA) + (BAB) = AB + BA for some/all s > 1. s h72i ( Im − AB ) = Im − AB for some/all s > 2. s h73i ( Im − AB ) = Im − BA for some/all s > 2. s s h74i ( Im − AB ) = ( Im − BA ) for some/all s > 1. s h75i ( A + B − AB ) = A + B − AB for some/all s > 2. h76i ( A + B − AB )† = A + B − AB. m h77i A + B − AB ∈ COP. 2s 2s−1 h78i ( AB + BA ) = 2 ( AB + BA ) for some/all s > 1. 2 h79i ( Im − AB − BA ) = Im . − − h80i Im + AB ∈ {( Im − AB ) } and/or Im + BA ∈ {( Im − BA ) }. s s h81i ( Im − ABA ) = Im − ABA and/or ( Im − BAB ) = Im − BAB for some/all s > 2. s s h82i ( Im − ABA ) + ( Im − BAB ) = 2Im − ABA − BAB for some/all s > 2. s s−1 h83i ( ABA + BAB ) = 2 ( ABA + BAB ) for some/all s > 2. 2 h84i ( Im − ABA − BAB ) = Im . s s h85i ( A − BAB ) = A − BAB and/or ( B − ABA ) = B − ABA for some/all s > 2. s s h86i ( A − BAB ) + ( B − ABA ) = A + B − ABA − BAB for some/all s > 2. h87i A − BAB < 0 and/or B − ABA < 0. h88i A + B − ABA − BAB < 0. m m h89i A − BAB ∈ COP and/or B − ABA ∈ COP. m h90i ( A + B − ABA − BAB )/2 ∈ COP. m m h91i ABA + ABe Ae ∈ COP and/or BAB + BAe Be ∈ COP. h92i ABA + ABe Ae = B and/or BAB + BAe Be = A. h93i ABAe + ABAe = 0 and/or BABe + BABe = 0, i.e., AB + BA = 2ABA and/or AB + BA = 2BAB. h94i (AB)† = BA and/or (BA)† = AB. h95i (ABe )† = BAe and/or (BAe)† = ABe . h96i (ABe)† = BAe and/or (BAe )† = ABe. h97i (AeBe)† = BeAe and/or (BeAe)† = AeBe. h98i (AB)† = AB and/or (BA)† = BA. h99i (ABe )† = ABe and/or (BAe)† = BAe. h100i (ABe)† = ABe and/or (BAe )† = BAe . h101i (AeBe)† = AeBe and/or (BeAe)† = BeAe. h102i Some/all of the four facts:(AB)† + (BA)† = AB + BA, (ABe )† + (BAe)† = ABe + BAe, (ABe)† + (BAe )† = ABe + BAe , (AeBe)† + (BeAe)† = AeBe + BeAe. h103i Some/all of the eight facts:[(AB)2 ]† = [ (AB)† ]2, [(BA)2 ]† = [ (BA)† ]2, [(ABe )2 ]† = [ (ABe )† ]2, [(BAe)2 ]† = [(BAe)† ]2, [(ABe)2 ]† = [ (ABe)† ]2, [(BAe )2 ]† = [ (BAe )† ]2, [(AeBe)2 ]† = [ (AeBe)† ]2, [(BeAe)2 ]† = [ (BeAe)† ]2. † † m † † m † † m h104i Some/all of the four facts:[(AB) +(BA) ]/2 ∈ COP, [(ABe ) +(BAe) ]/2 ∈ COP, [(ABe) +(BAe ) ]/2 ∈ COP, † † m [(AeBe) + (BeAe) ]/2 ∈ COP. Linear subspaces and orthogonal projectors Ë 189 h105i (AB)(AB)† = ABA and/or BA(BA)† = BAB. h106i (AB)(AB)† + (BA)(BA)† = ABA + BAB. h107i (AB)(BA)† = (BA)(AB)†. h108i [(AB)(AB)†][(BA)(BA)†] = [(BA)(BA)†][(AB)(AB)†]. h109i A(AeBe)†B = 0 and/or B(BeAe)†A = 0. h110i A(AeBe)†B + B(BeAe)†A = 0. # m # m h111i (AB) ∈ COP and/or (BA) ∈ COP. # # m h112i [(AB) + (BA) ]/2 ∈ COP. h113i (AB)# = (AB)† and/or (BA)# = (BA)†. h114i (AB)# + (BA)# = (AB)† + (BA)†. h115i (AB)(AB)# = (AB)(AB)† and/or (BA)(BA)# = (BA)(BA)†. h116i (AB)(AB)# + (BA)(BA)# = (AB)(AB)† + (BA)(BA)†. h117i (AB)# = AB and/or (BA)# = BA. h118i (AB)# + (BA)# = AB + BA. h119i (ABA)† = ABA and/or (BAB)† = BAB. h120i (ABA)† + (BAB)† = ABA + BAB. † † h121i ( Im − AB ) = Im − AB and/or ( Im − BA ) = Im − BA. † † h122i ( Im − AB ) = Im − BA and/or ( Im − BA ) = Im − AB. † † h123i ( Im − AB ) + ( Im − BA ) = 2Im − AB − BA. 2 † † 2 2 † † 2 h124i [( Im − AB ) ] = [ ( Im − AB ) ] and/or [( Im − BA ) ] = [ ( Im − BA ) ] . † † h125i A( Im − BA ) B = 0 and/or B( Im − AB ) A = 0. † † h126i A( Im − BA ) B + B( Im − AB ) A = 0. † † m h127i [( Im − AB ) + ( Im − BA ) ]/2 ∈ COP. † † h128i ( 2Im − A − B )( 2Im − A − B ) = Im − AB and/or ( 2Im − A − B )( 2Im − A − B ) = Im − BA. † † h129i ( 2Im − AB − BA )( 2Im − AB − BA ) = Im − AB and/or ( 2Im − AB − BA )( 2Im − AB − BA ) = Im − BA. † † h130i ( Im − ABA ) = Im − ABA and/or ( Im − BAB ) = Im − BAB. † † h131i ( Im − ABA ) + ( Im − BAB ) = 2Im − ABA − BAB. † † m h132i [(ABA) + (BAB) ]/2 ∈ COP. † † h133i A( Im − ABA ) B = 0 and/or A( Im − BAB ) B = 0. † † h134i A( Im − ABA ) B + A( Im − BAB ) B = 0. † † h135i ( 2Im − ABA − BAB )( 2Im − ABA − BAB ) = Im − ABA and/or ( 2Im − ABA − BAB )( 2Im − ABA − BAB ) = Im − BAB. s h136i Trace[ (AB) ] = Trace(AB) for some/all s > 2. h137i Some/all of the four facts: r(AB) = Trace(AB), r(ABe ) = Trace(ABe ), r(ABe) = Trace(ABe), r(AeBe) = Trace(AeBe). h138i Trace[( A − B )2] = r(A) + r(B) − 2r(AB). h139i Trace[( A − B )2] = Trace(A) + Trace(B) − 2r(AB). 2s 2 h140i Trace[( A − B ) ] = Trace[( A − B ) ] for some/all s > 2. s h141i Trace[ ( Im − AB ) ] = Trace( Im − AB ) for some/all s > 2. s s h142i Trace[ ( Im − ABA ) ] = Trace( Im − ABA ) and/or Trace[ ( Im − BAB ) ] = Trace( Im − BAB ) for some/all s > 2. h143i Some/all of the four facts: dim(M∩N) = r(AB), dim(M⊥ ∩N) = r(ABe ), dim(M∩N⊥) = r(ABe), dim(M⊥ ∩ N⊥) = r(AeBe). h144i Some/all of the four facts: r[ A, B ] = r(A) + r(B) − r(AB), r[ Ae, B ] = r(Ae) + r(B) − r(ABe ), r[ A, Be ] = r(A) + r(Be) − r(ABe), r[ Ae, Be ] = r(Ae) + r(Be) − r(AeBe). h145i Some/all of the four facts: r[ A, B ] = Trace(A) + Trace(B) − Trace(AB), r[ Ae, B ] = Trace(Ae) + Trace(B) − Trace(ABe ), r[ A, Be ] = Trace(A) + Trace(Be) − Trace(ABe), r[ Ae, Be ] = Trace(Ae) + Trace(Be) − Trace(AeBe). h146i r( A − B ) = r(A) + r(B) − 2r(AB). h147i r( A − B ) = Trace(A) + Trace(B) − 2Trace(AB). h148i r( Im − A − B ) = m − r(A) − r(B) + 2r(AB). h149i r( Im − A − B ) = m − Trace(A) − Trace(B) + 2Trace(AB). 190 Ë Yongge Tian

h150i r( A − B ) + r( Im − A − B ) = m. h151i 2r[ A, B ] = 2r(A) + 2r(B) − r( AB + BA ). h152i Some/all of the four facts: r( Im − AB ) = m − r(AB), r( Im − ABe ) = m − r(ABe ), r( Im − ABe ) = m − r(ABe), r( Im − AeBe ) = m − r(AeBe). h153i Some/all of the four facts: r( Im − AB ) = m − Trace(AB), r( Im − ABe ) = m − Trace(ABe ), r( Im − ABe ) = m − Trace(ABe), r( Im − AeBe ) = m − Trace(AeBe). h154i r( A − AB ) = r(A) − r(AB) and/or r( B − BA ) = r(B) − r(BA). h155i r( A − AB ) = Trace(A) − Trace(AB) and/or r( B − BA ) = Trace(B) − Trace(BA). h156i r( A + B − AB ) = r(A) + (B) − r(AB). h157i r( A + B − AB ) = Trace(A) + Trace(B) − Trace(AB). h158i r( Im − ABA ) = m − r(ABA) and/or r( Im − BAB ) = m − r(BAB). h159i r( Im − ABA ) = m − Trace(ABA) and/or r( Im − BAB ) = m − Trace(BAB). h160i r( A − BAB ) = r(A) − r(BAB) and/or r( B − ABA ) = r(B) − r(ABA). h161i r( A − BAB ) = Trace(A) − Trace(BAB) and/or r( B − ABA ) = Trace(B) − Trace(ABA). h162i A − B = PM∩N⊥ − PM⊥∩N. h163i R( A − B ) = (M ∩ N⊥) ⊕ (M⊥ ∩ N). h164i PR( A−B ) = PM∩N⊥ + PM⊥∩N. h165i Im − A − B = −PM∩N + PM⊥∩N⊥ . ⊥ ⊥ h166i R( Im − A − B ) = (M ∩ N) ⊕ (M ∩ N ). h i ⊥ ⊥ 167 PR( Im−A−B ) = PM∩N + PM ∩N . h168i R( A − B ) ∩ R( Im − A − B ) = {0}. m h169i C = R( A − B ) ⊕ R( Im − A − B ). h i 170 Im = PR( A−B ) + PR( Im−A−B ). h171i Some/all of the eight facts: AB = PM∩N and/or ABe = PM⊥∩N, ABe = PM∩N⊥ , AeBe = PM⊥∩N⊥ , BA = PM∩N, BAe = PM⊥∩N, BAe = PM∩N⊥ , BeAe = PM⊥∩N⊥ . h172i Some/all of the four facts: AB + BA = 2PM∩N, ABe + BAe = 2PM⊥∩N, ABe + BAe = 2PM∩N⊥ , AeBe + BeAe = 2PM⊥∩N⊥ . † † h173i AB = PR(AB) and/or BA = PR(BA), i.e., AB = (AB)(AB) and/or BA = (BA)(BA) . h174i AB = P and/or BA = P , i.e., AB = (AB)(AB)† and/or BA = (BA)(BA)†. e R(eAB) e R(BeA) e e e e e e h175i AB = P and/or BA = P , i.e., AB = (AB)(AB)† and/or BA = (BA)(BA)†. e R(AeB) e R(eBA) e e e e e e h176i AB = P and/or BA = P , i.e., AB = (AB)(AB)† and/or BA = (BA)(BA)†. ee R(eAeB) ee R(eBeA) ee ee ee ee ee ee h177i AB + BA = PR(AB) + PR(BA). h178i Some/all of the four facts: R(AB) = M ∩ N, R(ABe ) = M⊥ ∩ N, R(ABe) = M ∩ N⊥, R(AeBe) = M⊥ ∩ N⊥. h179i Some/all of the four facts: R( AB + BA ) = M ∩ N, R( ABe + BAe ) = M⊥ ∩ N, R( ABe + BAe ) = M ∩ N⊥, R( AeBe + BeAe ) = M⊥ ∩ N⊥. h180i A − AB = PM∩N⊥ and/or B − BA = PM⊥∩N. 2 h181i ( A − B ) = PM∩N⊥ + PM⊥∩N. h182i A + B − AB = PM∩N + PM∩N⊥ + PM⊥∩N. ⊥ ⊥ ⊥ ⊥ h183i R( Im − AB ) = R( Im − BA ) = (M ∩ N ) ⊕ (M ∩ N) ⊕ (M ∩ N ). h184i Im − AB = PM∩N⊥ + PM⊥∩N + PM⊥∩N⊥ . h185i R(AB) ∩ R( Im − AB ) = {0} and/or R(BA) ∩ R( Im − BA ) = {0}. h186i R(ABe ) ∩ R( Im − ABe ) = {0} and/or R(BAe) ∩ R( Im − BAe ) = {0}. h187i R(ABe) ∩ R( Im − ABe ) = {0} and/or R(BAe ) ∩ R( Im − BAe ) = {0}. h188i R(AeBe) ∩ R( Im − AeBe ) = {0} and/or R(BeAe) ∩ R( Im − BeAe ) = {0}. m m h189i C = R(AB) ⊕ R( Im − AB ) and/or C = R(BA) ⊕ R( Im − BA ). m m h190i C = R(ABe ) ⊕ R( Im − ABe ) and/or C = R(BAe) ⊕ R( Im − BAe ). m m h191i C = R(ABe) ⊕ R( Im − ABe ) and/or C = R(BAe ) ⊕ R( Im − BAe ). m m h192i C = R(AeBe) ⊕ R( Im − AeBe ) and/or C = R(BeAe) ⊕ R( Im − BeAe ). h i 193 Im = PR(AB) + PR( Im−AB ) and/or Im = PR(BA) + PR( Im−BA ). h194i ABA = PM∩N and/or BAB = PM∩N. Linear subspaces and orthogonal projectors Ë 191

h195i ABe Ae = PM⊥∩N and/or BABe = PM⊥∩N. h196i ABAe = PM∩N⊥ and/or BAe Be = PM∩N⊥ . h197i AeBeAe = PM⊥∩N⊥ and/or BeAeBe = PM⊥∩N⊥ . h198i ABA + BAB = 2PM∩N. h199i ABe Ae + BABe = 2PM⊥∩N. h200i ABAe + BAe Be = 2PM∩N⊥ . h201i AeBeAe + BeAeBe = 2PM⊥∩N⊥ . h202i ABA + ABe Ae = PM∩N + PM⊥∩N and/or BAB + BAe Be = PM∩N + PM∩N⊥ . h203i A − BAB = PM∩N⊥ and/or B − ABA = PM⊥∩N. h204i R( A − BAB ) = M ∩ N⊥ and/or R( B − ABA ) = M⊥ ∩ N. h205i Some/all of the four facts: R(AB) = R(BA), R(ABe ) = R(BAe), R(ABe) = R(BAe ), R(AeBe) = R(BeAe). h206i Some/all of the four facts: N (AB) = N (BA), N (ABe ) = N (BAe), N (ABe) = N (BAe ), N (AeBe) = N (BeAe), m m h207i R(AB) ⊕ N (AB) = C and/or R(BA) ⊕ N (BA) = C . m m h208i R(ABe ) ⊕ N (ABe ) = C and/or R(BAe) ⊕ N (BAe) = C . m m h209i R(ABe) ⊕ N (ABe) = C and/or R(BAe ) ⊕ N (BAe ) = C . m m h210i R(AeBe) ⊕ N (AeBe) = C and/or R(BeAe) ⊕ N (BeAe) = C . h211i R(AB) ⊆ R(B) and/or R(BA) ⊆ R(A). h212i R(ABe ) ⊆ R(B) and/or R(BAe) ⊆ R(A). h213i R(ABe) ⊆ R(Be) and/or R(BAe ) ⊆ R(A). h214i R(AeBe) ⊆ R(Be) and/or R(BeAe) ⊆ R(Ae). h215i R(AB) = R(A) ∩ R(B) and/or R(BA) = R(A) ∩ R(B). h216i R(ABe ) = R(Ae) ∩ R(B) and/or R(BAe) = R(Ae) ∩ R(B). h217i R(ABe) = R(A) ∩ R(Ae) and/or R(BAe ) = R(A) ∩ R(Be). h218i R(AeBe) = R(Ae) ∩ R(Be) and/or R(BeAe) = R(Ae) ∩ R(Be). h219i R(ABAe) = R(BABe). h220i N (ABAe) = N (BABe).

Proof. The equivalences of h26i–h112i and h121i–h220i follow from some of (3.3), (3.4), Theorem 3.4(a), and (4.93)–(4.111). The equivalences of h29i and h113i–h118i follow from the last ve equalities in Theorem 3.4(d). It is easy to verify that

AB = BA ⇔ ABe = BAe ⇔ ABe = BAe ⇔ AeBe = BeAe for two orthogonal projectors. In consequence, replacing A and B with Ae and Be in Theorem 4.1 respectively will yield other three groups of equivalent statements for the above statements to hold. Note that A ⊗ B = (A ⊗ Im)(Im ⊗ B) = (Im ⊗ B)(A ⊗ Im).

Applying Theorem 5.2 to the commutative orthogonal projectors A ⊗ Im and Im ⊗ B and noting that r(A ⊗ B) = r(A)r(B), r(A ⊗ Im) = mr(A) and r(Im ⊗ B) = mr(B), we obtain the following result.

m Corollary 5.3. Let M and N be two linear subspaces of C with A = PM and B = PN, and let

k3 = r(A)r(B), k4 = mr(A) − r(A)r(B), k5 = mr(B) − r(A)r(B), 2 k6 = m − mr(A) − mr(B) + r(A)r(B).

Then, there exists a unitary matrix U such that

∗ ∗ A ⊗ Im = Udiag Ik3+k4 , 0k5 , 0k6 U , Im ⊗ B = Udiag Ik3 , 0k4 , Ik5 , 0k6 U . ∗ ∗ A ⊗ Im + Im ⊗ B = Udiag 2Ik3 , Ik4+k5 , 0k6 U , A ⊗ Im − Im ⊗ B = Udiag 0k3 , Ik4 , −Ik5 , 0k6 U , ∗ ⊗ 2 A B = Udiag Ik3 , 0m −k3 U . 192 Ë Yongge Tian

6 More on subspace inclusions and subspace equalities

As a continuation of Corollary 2.15, we reconsider in this section subspace inequalities and subspace identities, and present many well-known and new statements for subspace inequalities and subspace identities to hold.

m Theorem 6.1. Let M and N be two linear subspaces of C , A, B, Ae and Be be as given in (3.2), let s be an integer, and let k1, ... , k6 be as given in (3.8) and (3.9). Then, the following statements are equivalent: h1i M ⊇ N. h2i–h37i in Corollary 2.15. h38i M = R(A − B) ⊕ N. h39i N⊥ = M⊥ ⊕ R(A − B). h40i M + N⊥ = (M⊥ + N) ⊕ R(A − B). h41i AB = BA = B. h42i AB + BA = 2B. h43i ABA = B. h44i BAB = B. h45i ABA + BAB = 2B. s s h46i (AB) = (BA) = B for some/all s > 2. s s h47i (AB) + (BA) = 2B for some/all s > 2. s h48i (ABA) = B for some/all s > 2. s h49i (BAB) = B for some/all s > 2. s s h50i (ABA) + (BAB) = 2B for some/all s > 2. h51i BeAe = BeAe = Ae. h52i BeAe + BeAe = 2Ae. h53i BeAeBe = Ae. h54i AeBeAe = Ae. h55i BeAeBe + AeBeAe = 2Ae. s s h56i (BeAe) = (BeAe) = Ae for some/all s > 2. s s h57i (BeAe) + (BeAe) = 2Ae for some/all s > 2. s h58i (BeAeBe) = Ae for some/all s > 2. s h59i (AeBeAe) = Ae for some/all s > 2. s s h60i (BeAeBe) + (AeBeAe) = 2Ae for some/all s > 2. h61i A < B and/or Be < Ae. m m h62i A − B ∈ COP and/or Be − Ae ∈ COP. h63i r[ A, B ] = r(A) and/or r[ Ae, Be ] = r(Be). h64i A = PM∩N⊥ + B and/or Be = PM∩N⊥ + Ae. h65i B = PM∩N. h66i Ae = PM⊥∩N⊥ . h67i There exists a unitary matrix U such that

∗ ∗ A = Udiag Ik3 , Ik4 , 0k6 U and B = Udiag Ik3 , 0k4+k6 U . h68i There exists a unitary matrix U such that

∗ αA + βB = Udiag (α + β)Ik3 , αIk4 , 0k6 U

for some/all two nonzero real numbers α and β. h69i r(AB) = r(B) with one of the four facts: AB = BA, ABe = BAe , ABe = BAe, AeBe = BeAe. h70i r(AB) = r(B) with one of the four facts: ABA = BAB, ABAe = BAe Be, ABe Ae = BABe , AeBeAe = BeAeBe. h71i r(AB) = r(B) and A = PM∩N + PM∩N⊥ . h72i r(AB) = r(B) and A + B = 2PM∩N + PM∩N⊥ . Linear subspaces and orthogonal projectors Ë 193

h73i r(AB) = r(B) and Be = PM∩N⊥ + PM⊥∩N⊥ . h74i r(AB) = r(B) and Ae + Be = PM∩N⊥ + 2PM⊥∩N⊥ . h75i r(AB) = r(B) and AB + BA < 0. 2 2 h76i r(AB) = r(B) with one/both of the two facts:( A − B ) < A − B, ( B − A ) < B − A. 2 m h77i r(AB) = r(B) and ( A − B ) ∈ COP. 2s+1 h78i r(AB) = r(B) and ( A − B ) = A − B for some/all s > 1. h79i r(AB) = r(B) and ( A − B )† = A − B. h80i r(AB) = r(B) with one/both of the two facts: A( A − B )† = ( A − B )†A, B( A − B )† = ( A − B )†B. h81i r(AB) = r(B) with one/both of the two facts: A( A − B )† = ( A − B )†A, B( A − B )† = ( A − B )†B. h82i r(AB) = r(B) and ( A + B )( A − B )† = ( A − B )†( A + B ). h83i r(AB) = r(B) and A( A + B )†A + B( A + B )†B is an orthogonal projector. h84i r(AB) = r(B) with one/both of the two facts: A( A + B )† = ( A + B )†A, B( A + B )† = ( A + B )†B. 2 2 h85i r(AB) = r(B) with one/both of the two facts:( Im − A − B ) < Im − A − B, ( A + B − Im ) < A + B − Im . h86i r(AB) = r(B) and ( I − A − B )2 is an orthogonal projector. 2s+1 h87i r(AB) = r(B) and ( Im − A − B ) = Im − A − B for some/all s > 1. † h88i r(AB) = r(B) and ( Im − A − B ) = Im − A − B. † † h89i r(AB) = r(B) and ( A − B )( Im − A − B ) = ( Im − A − B ) ( A − B ). h90i r(AB) = r(B) and ( A + B )( A + B )† = A + B − AB. h91i r(AB) = r(B) and [( A + B )( A + B )† − A ][ ( A + B )( A + B )† − B ] = 0. h92i r(AB) = r(B) and ( A + B )( A + B )† = A + ABe Ae and/or ( A + B )( A + B )† = B + BAe Be. h93i r(AB) = r(B) and A + B = ( A − B )( A − B )† + 2(AB)(AB)†. † † h94i r(AB) = r(B) and A < (BA)(BA) and/or B < (AB)(AB) . † † h95i r(AB) = r(B) and A + B < (AB)(AB) + (BA)(BA) . † † h96i r(AB) = r(B) with one/both of the two facts: A < (BAe )(BAe ) , B < (ABe )(ABe ) . † † h97i r(AB) = r(B) and A + B < (ABe )(ABe ) + (BAe )(BAe ) . h98i r(AB) = r(B) and 2A( AB + BA )†B = AB and/or 2B( AB + BA )†A = BA. h99i r(AB) = r(B) and 2A( AB + BA )†B + 2B( AB + BA )†A = AB + BA. s m h100i r(AB) = r(B) and (AB) ∈ COP for some/all s > 2. s h101i r(AB) = r(B) and (AB) = AB for some/all s > 2. s s h102i r(AB) = r(B) and (AB) = (BA) for some/all s > 1. s h103i r(AB) = r(B) and (AB) = BA for some/all s > 2. s s h104i r(AB) = r(B) and (AB) + (BA) = AB + BA for some/all s > 2. s m s m h105i r(AB) = r(B) with one/both of the two facts:(ABA) ∈ COP, (BAB) ∈ COP for some/all s > 2. s s h106i r(AB) = r(B) with one/both of the two facts:(ABA) = ABA, (BAB) = BAB for some/all s > 2. s s h107i r(AB) = r(B) with one/both of the two facts:(ABA) = BAB, (BAB) = ABA for some/all s > 2. s s h108i r(AB) = r(B) and (ABA) + (BAB) = ABA + BAB for some/all s > 2. s s h109i r(AB) = r(B) with one/both of the two facts:(ABA) = AB, (BAB) = BA for some/all s > 1. s s h110i r(AB) = r(B) and (ABA) + (BAB) = AB + BA for some/all s > 1. s h111i r(AB) = r(B) and ( Im − AB ) = Im − AB for some/all s > 2. s h112i r(AB) = r(B) and ( Im − AB ) = Im − BA for some/all s > 2. s s h113i r(AB) = r(B) and ( Im − AB ) = ( Im − BA ) for some/all s > 1. s h114i r(AB) = r(B) and ( A + B − AB ) = A + B − AB for some/all s > 2. h115i r(AB) = r(B) and ( A + B − AB )† = A + B − AB. m h116i r(AB) = r(B) and A + B − AB ∈ COP. 2s 2s−1 h117i r(AB) = r(B) and ( AB + BA ) = 2 ( AB + BA ) for some/all s > 1. 2 h118i r(AB) = r(B) and ( Im − AB − BA ) = Im . − − h119i r(AB) = r(B) with one/both of the two facts: Im + AB ∈ {( Im − AB ) }, Im + BA ∈ {( Im − BA ) }. s s h120i r(AB) = r(B) with one/both of the two facts:( Im − ABA ) = Im − ABA, ( Im − BAB ) = Im − BAB for some/all s > 2. s s h121i r(AB) = r(B) and ( Im − ABA ) + ( Im − BAB ) = 2Im − ABA − BAB for some/all s > 2. s s−1 h122i r(AB) = r(B) and ( ABA + BAB ) = 2 ( ABA + BAB ) for some/all s > 2. 194 Ë Yongge Tian

2 h123i r(AB) = r(B) and ( Im − ABA − BAB ) = Im . h124i r(AB) = r(B) with one/both of the two facts:( A − BAB )s = A − BAB, ( B − ABA )s = B − ABA for some/all s > 2. s s h125i r(AB) = r(B) and ( A − BAB ) + ( B − ABA ) = A + B − ABA − BAB for some/all s > 2. h126i r(AB) = r(B) and A − BAB < 0 and/or B − ABA < 0. h127i r(AB) = r(B) and A + B − ABA − BAB < 0. m m h128i r(AB) = r(B) with one/both of the two facts: A − BAB ∈ COP, B − ABA ∈ COP. m h129i r(AB) = r(B) and ( A + B − ABA − BAB )/2 ∈ COP. m m h130i r(AB) = r(B) with one/both of the two facts: ABA + ABe Ae ∈ COP, BAB + BAe Be ∈ COP. h131i r(AB) = r(B) with one/both of the two facts: ABA + ABe Ae = B, BAB + BAe Be = A. h132i r(AB) = r(B) with one/both of the two facts: AB + BA = 2ABA, AB + BA = 2BAB. h133i r(AB) = r(B) with one/both of the two facts:(AB)† = BA, (BA)† = AB. h134i r(AB) = r(B) with one/both of the two facts:(AB)† = AB, (BA)† = BA. h135i r(AB) = r(B) and (AB)† + (BA)† = AB + BA. h136i r(AB) = r(B) and [(AB)2 ]† = [ (AB)† ]2. † † m h137i r(AB) = r(B) and [(AB) + (BA) ]/2 ∈ COP. h138i r(AB) = r(B) with one/both of the two facts: AB(AB)† = ABA, BA(BA)† = BAB. h139i r(AB) = r(B) and AB(AB)† + BA(BA)† = ABA + BAB. h140i r(AB) = r(B) and (AB)(AB)† = (BA)(BA)†. h141i r(AB) = r(B) and [(AB)(AB)†][(BA)(BA)†] = [(BA)(BA)†][(AB)(AB)†]. h142i r(AB) = r(B) with one/both of the two facts: A(AeBe)†B = 0, B(BeAe)†A = 0. h143i r(AB) = r(B) and A(AeBe)†B + B(BeAe)†A = 0. # m # m h144i r(AB) = r(B) with one/both of the two facts:(AB) ∈ COP, (BA) ∈ COP. # # m h145i r(AB) = r(B) and [(AB) + (BA) ]/2 ∈ COP. h146i r(AB) = r(B) with one/both of the two facts:(AB)# = (AB)†, (BA)# = (BA)†. h147i r(AB) = r(B) and (AB)# + (BA)# = (AB)† + (BA)†. h148i r(AB) = r(B) with one/both of the two facts:(AB)(AB)# = (AB)(AB)†, (BA)(BA)# = (BA)(BA)†. h149i r(AB) = r(B) and (AB)(AB)# + (BA)(BA)# = (AB)(AB)† + (BA)(BA)†. h150i r(AB) = r(B) and (AB)# = AB and/or (BA)# = BA. h151i r(AB) = r(B) and (AB)# + (BA)# = AB + BA. h152i r(AB) = r(B) with one/both of the two facts:(ABA)† = ABA, (BAB)† = BAB. h153i r(AB) = r(B) and (ABA)† + (BAB)† = ABA + BAB. † † h154i r(AB) = r(B) with one/both of the two facts:( Im − AB ) = Im − AB, ( Im − BA ) = Im − BA. † † h155i r(AB) = r(B) with one/both of the two facts:( Im − AB ) = Im − BA, ( Im − BA ) = Im − AB. † † h156i r(AB) = r(B) and ( Im − AB ) + ( Im − BA ) = 2Im − AB − BA. 2 † † 2 h157i r(AB) = r(B) and [( Im − AB ) ] = [ ( Im − AB ) ] . † † h158i r(AB) = r(B) with one/both of the two facts: A( Im − BA ) B = 0, B( Im − AB ) A = 0. † † h159i r(AB) = r(B) and A( Im − BA ) B + B( Im − AB ) A = 0. † † m h160i r(AB) = r(B) and [( Im − AB ) + ( Im − BA ) ]/2 ∈ COP. † h161i r(AB) = r(B) with one/both of the two facts: ( 2Im − A − B )( 2Im − A − B ) = Im − AB, ( 2Im − A − B )( 2Im − † A − B ) = Im − BA. † h162i r(AB) = r(B) with one/both of the two facts: ( 2Im − AB − BA )( 2Im − AB − BA ) = Im − AB, ( 2Im − AB − † BA )( 2Im − AB − BA ) = Im − BA. † † h163i r(AB) = r(B) with one/both of the two facts:( Im − ABA ) = Im − ABA, ( Im − BAB ) = Im − BAB. † † h164i r(AB) = r(B) and ( Im − ABA ) + ( Im − BAB ) = 2Im − ABA − BAB. † † m h165i r(AB) = r(B) and [(ABA) + (BAB) ]/2 ∈ COP. † † h166i r(AB) = r(B) with one/both of the two facts: A( Im − ABA ) B = 0, A( Im − BAB ) B = 0. † † h167i r(AB) = r(B) and A( Im − ABA ) B + A( Im − BAB ) B = 0. † h168i r(AB) = r(B) with one/both of the two facts: ( 2Im − ABA − BAB )( 2Im − ABA − BAB ) = Im − ABA, † ( 2Im − ABA − BAB )( 2Im − ABA − BAB ) = Im − BAB. s h169i r(AB) = r(B) and Trace[ (AB) ] = Trace(AB) for some/all s > 2. Linear subspaces and orthogonal projectors Ë 195 h170i r(AB) = r(B) and r(AB) = Trace(AB). h171i r(AB) = r(B) and Trace[( A − B )2] = r(A) + r(B) − 2r(AB). h172i r(AB) = r(B) and Trace[( A − B )2] = Trace(A) + Trace(B) − 2r(AB). 2s 2 h173i r(AB) = r(B) and Trace[( A − B ) ] = Trace[( A − B ) ] for some/all s > 2. s h174i r(AB) = r(B) and Trace[ ( Im − AB ) ] = Trace( Im − AB ) for some/all s > 2. s s h175i r(AB) = r(B) and Trace[ ( Im −ABA ) ] = Trace( Im −ABA ) and/or Trace[ ( Im −BAB ) ] = Trace( Im −BAB ) for some/all s > 2. h176i r(AB) = r(B) and dim(M ∩ N) = r(AB). h177i r(AB) = r(B) and r[ A, B ] = r(A) + r(B) − r(AB). h178i r(AB) = r(B) and r[ A, B ] = Trace(A) + Trace(B) − Trace(AB). h179i r(AB) = r(B) and r( A − B ) = r(A) + r(B) − 2r(AB). h180i r(AB) = r(B) and r( A − B ) = Trace(A) + Trace(B) − 2Trace(AB). h181i r(AB) = r(B) and r( Im − A − B ) = m − r(A) − r(B) + 2r(AB). h182i r(AB) = r(B) and r( Im − A − B ) = m − Trace(A) − Trace(B) + 2Trace(AB). h183i r(AB) = r(B) and r( A − B ) + r( Im − A − B ) = m. h184i r(AB) = r(B) and 2r[ A, B ] = 2r(A) + 2r(B) − r( AB + BA ). h185i r(AB) = r(B) and r( Im − AB ) = m − r(AB). h186i r(AB) = r(B) and r( Im − AB ) = m − Trace(AB). h187i r(AB) = r(B) and r( A − AB ) = r(A) − r(AB). h188i r(AB) = r(B) and r( A − AB ) = Trace(A) − Trace(AB). h189i r(AB) = r(B) and r( A + B − AB ) = r(A) + (B) − r(AB). h190i r(AB) = r(B) and r( A + B − AB ) = Trace(A) + Trace(B) − Trace(AB). h191i r(AB) = r(B) with one/both of the two facts: r( Im − ABA ) = m − r(ABA), r( Im − BAB ) = m − r(BAB). h192i r(AB) = r(B) with one/both of the two facts: r( Im − ABA ) = m − Trace(ABA), r( Im − BAB ) = m − Trace(BAB). h193i r(AB) = r(B) with one/both of the two facts: r( A − BAB ) = r(A) − r(BAB), r( B − ABA ) = r(B) − r(ABA). h194i r(AB) = r(B) with one/both of the two facts: r( A − BAB ) = Trace(A) − Trace(BAB), r( B − ABA ) = Trace(B) − Trace(ABA). h195i r(AB) = r(B) and R( A − B ) = M ∩ N⊥. h196i r(AB) = r(B) and PR( A−B ) = PM∩N⊥ . h197i r(AB) = r(B) and Im − A − B = −PM∩N + PM⊥∩N⊥ . ⊥ ⊥ h198i r(AB) = r(B) and R( Im − A − B ) = (M ∩ N) ⊕ (M ∩ N ). h i ⊥ ⊥ 199 r(AB) = r(B) and PR( Im−A−B ) = PM∩N + PM ∩N . h200i r(AB) = r(B) and R( A − B ) ∩ R( Im − A − B ) = {0}. m h201i r(AB) = r(B) and C = R( A − B ) ⊕ R( Im − A − B ). h i 202 r(AB) = r(B) and Im = PR( A−B ) + PR( Im−A−B ). h203i r(AB) = r(B) with one/both of the two facts: AB = PM∩N, BA = PM∩N. h204i r(AB) = r(B) and AB + BA = 2PM∩N. h205i r(AB) = r(B) with one/both of the two facts: AB = (AB)(AB)†, BA = (BA)(BA)†. h206i r(AB) = r(B) and AB + BA = PR(AB) + PR(BA). h207i r(AB) = r(B) with one/both of the two facts: R(AB) = M ∩ N, R(BA) = M ∩ N. h208i r(AB) = r(B) and R( AB + BA ) = M ∩ N. h209i r(AB) = r(B) and A − AB = PM∩N⊥ . 2 h210i r(AB) = r(B) and ( A − B ) = PM∩N⊥ . h211i r(AB) = r(B) and A + B − AB = PM∩N + PM∩N⊥ . ⊥ ⊥ ⊥ h212i r(AB) = r(B) and R( Im − AB ) = R( Im − BA ) = (M ∩ N ) ⊕ (M ∩ N ). h213i r(AB) = r(B) and Im − AB = PM∩N⊥ + PM⊥∩N⊥ . h214i r(AB) = r(B) with one/both of the two facts: R(AB) ∩ R( Im − AB ) = {0}, R(BA) ∩ R( Im − BA ) = {0}. m m h215i r(AB) = r(B) with one/both of the two facts: C = R(AB) ⊕ R( Im − AB ), C = R(BA) ⊕ R( Im − BA ). h i 216 r(AB) = r(B) with one/both of the two facts: Im = PR(AB) + PR( Im−AB ), Im = PR(BA) + PR( Im−BA ). h217i r(AB) = r(B) with one/both of the two facts: ABA = PM∩N, BAB = PM∩N. 196 Ë Yongge Tian

h218i r(AB) = r(B) and ABA + BAB = 2PM∩N. h219i r(AB) = r(B) with one/both of the two facts: ABA + ABe Ae = PM∩N, BAB + BAe Be = PM∩N + PM∩N⊥ . h220i r(AB) = r(B) and A − BAB = PM∩N⊥ . h221i R( A − BAB ) = M ∩ N⊥. h222i r(AB) = r(B) and R(AB) = R(BA). h223i r(AB) = r(B) and N (AB) = N (BA). m m h224i r(AB) = r(B) and R(AB) ⊕ N (AB) = C and/or R(BA) ⊕ N (BA) = C . h225i r(AB) = r(B) with one/both of the two facts: R(AB) ⊆ R(B), R(BA) ⊆ R(A). h226i r(AB) = r(B) with one/both of the two facts: R(AB) = R(A) ∩ R(B), R(BA) = R(A) ∩ R(B). h227i r(AB) = r(B) with one/both of the two facts: R(AeBe) = N (A) ∩ N (B), R(BeAe) = N (A) ∩ N (B). h228i r(AB) = r(B) with one/both of the two facts: R(ABAe) = R(BABe), r(AB) = r(B) and N (ABAe) = N (BABe).

m Theorem 6.2. Let M and N be two linear subspaces of C , A, B, Ae and Be be as given in (3.2), s be an integer, and k1, ... , k6 be as given in (3.8) and (3.9). Then, the following statements are equivalent: h1i M = N. h2i–h31i in Corollary 2.16. h32i A = B. h33i AB = B and BA = A. h34i AB + BA = A + B. s s h35i (AB) = B and (BA) = A for some/all s > 2. s s h36i (AB) + (BA) = A + B for some/all s > 2. h37i ABA = A and BAB = B. h38i ABA = B and BAB = A. h39i ABA + BAB = A + B. s s h40i (ABA) = A and (BAB) = B for some/all s > 2. s s h41i (ABA) = B and (BAB) = A for some/all s > 2. s s h42i (ABA) + (BAB) = A + B for some/all s > 2. h43i Ae = Be. h44i AeBe = Be and BeAe = Ae. h45i AeBe + BeAe = Ae + Be. s s h46i (AeBe) = Be and (BeAe) = Ae for some/all s > 2. s s h47i (AeBe) + (BeAe) = Ae + Be for some/all s > 2. h48i AeBeAe = Be and BeAeBe = Ae. h49i AeBeAe = Ae and BeAeBe = Be. h50i AeBeAe + BeAeBe = Ae + Be. s s h51i (AeBeAe) = Be and (BeAeBe) = Ae for some/all s > 2. s s h52i (AeBeAe) = Ae and (BeAeBe) = Be for some/all s > 2. s s h53i (AeBeAe) + (BeAeBe) = Ae + Be for some/all s > 2. h54i M = R(A − B) ⊕ N and N = R(A − B) ⊕ M. h55i M⊥ = R(A − B) ⊕ N⊥ and N⊥ = M⊥ ⊕ R(A − B). h56i 2r[ A, B ] = r(A) + r(B). h57i r[ A, B ] = r(A) = r(B). h58i r[ A, B ] = r(AB). h59i 2r[ Ae, Be ] = r(Ae) + r(Be). h60i r[ Ae, Be ] = r(Ae) = r(Be). h61i r[ Ae, Be ] = r(AeBe). h62i There exists a unitary matrix U such that

∗ αA + βB = Udiag (α + β)Ik3 , 0k6 U

for some/all two nonzero real numbers α and β. Linear subspaces and orthogonal projectors Ë 197 h63i There exists a unitary matrix U such that

∗ αAB + βBA = Udiag (α + β)Ik3 , 0m−k3 U

for some/all two nonzero real numbers α and β. h64i There exists a unitary matrix U such that

∗ αABA + βBAB = Udiag (α + β)Ik3 , 0m−k3 U

for some/all two nonzero real numbers α and β. h65i There exists a unitary matrix U such that

∗ αA + βBAB = Udiag (α + β)Ik3 , 0m−k3 U ,

for some/all two nonzero real numbers α and β, and/or there exists a unitary matrix U such that

∗ αB + βABA = Udiag (α + β)Ik3 , 0m−k3 U

for some/all two nonzero real numbers α and β. h66i There exists a unitary matrix U such that

† † ∗ α(AB) + β(BA) = Udiag (α + β)Ik3 , 0m−k3 U

for some/all two nonzero real numbers α and β. h67i There exists a unitary matrix U such that

† † ∗ α(ABA) + β(BAB) = Udiag (α + β)Ik3 , 0m−k3 U

for some/all two nonzero real numbers α and β. h68i r(AB) = r(A) = r(B) with some/all of the four facts: AB = BA, ABe = BAe , ABe = BAe, AeBe = BeAe. h69i r(AB) = r(A) = r(B) with some/all of the four facts: ABA = BAB, ABAe = BAe Be, ABe Ae = BABe , AeBeAe = BeAeBe. h70i r(AB) = r(A) = r(B) with one/both of the two facts: AB + BA 4 A, AB + BA 4 B. h71i r(AB) = r(A) = r(B) and AB + BA < 0. † † m h72i r(AB) = r(A) = r(B) and A( A + B ) A + B( A + B ) B ∈ COP. h73i r(AB) = r(A) = r(B) with one/both of the two facts: A( A + B )† = ( A + B )†A, B( A + B )† = ( A + B )†B. 2 2 h74i r(AB) = r(A) = r(B) with one/both of ( Im − A − B ) < Im − A − B, ( A + B − Im ) < A + B − Im . 2 m h75i r(AB) = r(A) = r(B) and ( I − A − B ) ∈ COP. 2s+1 h76i r(AB) = r(A) = r(B) and ( Im − A − B ) = Im − A − B for some/all s > 1. † h77i r(AB) = r(A) = r(B) and ( Im − A − B ) = Im − A − B. h78i r(AB) = r(A) = r(B) and ( A + B )( A + B )† = A + B − AB. h79i r(AB) = r(A) = r(B) and [( A + B )( A + B )† − A ][ ( A + B )( A + B )† − B ] = 0. h80i r(AB) = r(A) = r(B) with one/both of the two facts:( A+B )( A+B )† = A+ABe Ae, ( A+B )( A+B )† = B+BAe Be. h81i r(AB) = r(A) = r(B) and ( A + B )( A + B )† = (AB)(AB)†. † † h82i r(AB) = r(A) = r(B) with one/both of the two facts: A < (BA)(BA) , B < (AB)(AB) . † † h83i r(AB) = r(A) = r(B) and A + B < (AB)(AB) + (BA)(BA) . † † h84i r(AB) = r(A) = r(B) with one/both of the two facts: A < (BAe )(BAe ) , B < (ABe )(ABe ) . † † h85i r(AB) = r(A) = r(B) and A + B < (ABe )(ABe ) + (BAe )(BAe ) . h86i r(AB) = r(A) = r(B) with one/both of the two facts: 2A( AB + BA )†B = AB, 2B( AB + BA )†A = BA. h97i r(AB) = r(A) = r(B) and 2A( AB + BA )†B + 2B( AB + BA )†A = AB + BA. s m h88i r(AB) = r(A) = r(B) and (AB) ∈ COP for some/all s > 1. s h89i r(AB) = r(A) = r(B) and (AB) = AB for some/all s > 2. s s h90i r(AB) = r(A) = r(B) and (AB) = (BA) for some/all s > 1. s h91i r(AB) = r(A) = r(B) and (AB) = BA for some/all s > 2. s s h92i r(AB) = r(A) = r(B) and (AB) + (BA) = AB + BA for some/all s > 2. s m s m h93i r(AB) = r(A) = r(B) with one/both of the two facts:(ABA) ∈ COP, (BAB) ∈ COP for some/all s > 1. 198 Ë Yongge Tian

s s h94i r(AB) = r(A) = r(B) with one/both of the two facts:(ABA) = ABA, (BAB) = BAB for some/all s > 2. s s h95i r(AB) = r(A) = r(B) with some/all of (ABA) = BAB, (BAB) = ABA for some/all s > 1. s s h96i r(AB) = r(A) = r(B) and (ABA) + (BAB) = ABA + BAB for some/all s > 2. s s h97i r(AB) = r(A) = r(B) with one/both of the two facts:(ABA) = AB, (BAB) = BA for some/all s > 1. s s h98i r(AB) = r(A) = r(B) and (ABA) + (BAB) = AB + BA for some/all s > 1. s h99i r(AB) = r(A) = r(B) and ( Im − AB ) = Im − AB for some/all s > 2. s h100i r(AB) = r(A) = r(B) and ( Im − AB ) = Im − BA for some/all s > 2. s s h101i r(AB) = r(A) = r(B) and ( Im − AB ) = ( Im − BA ) for some/all s > 1. s h102i r(AB) = r(A) = r(B) and ( A + B − AB ) = A + B − AB for some/all s > 2. h103i r(AB) = r(A) = r(B) and ( A + B − AB )† = A + B − AB. m h104i r(AB) = r(A) = r(B) and A + B − AB ∈ COP. 2s 2s−1 h105i r(AB) = r(A) = r(B) and ( AB + BA ) = 2 ( AB + BA ) for some/all s > 1. 2 h106i r(AB) = r(A) = r(B) and ( Im − AB − BA ) = Im . − − h107i r(AB) = r(A) = r(B) with one/both of the two facts Im + AB ∈ {( Im − AB ) }, Im + BA ∈ {( Im − BA ) }. s s h108i r(AB) = r(A) = r(B) with one/both of the two facts:( Im − ABA ) = Im − ABA, ( Im − BAB ) = Im − BAB, for some/all s > 2. s s h109i r(AB) = r(A) = r(B) and ( Im − ABA ) + ( Im − BAB ) = 2Im − ABA − BAB for some/all s > 2. s s−1 h110i r(AB) = r(A) = r(B) and ( ABA + BAB ) = 2 ( ABA + BAB ) for some/all s > 2. 2 h111i r(AB) = r(A) = r(B) and ( Im − ABA − BAB ) = Im . h112i r(AB) = r(A) = r(B) with one/both of the two facts:( A − BAB )s = A − BAB, ( B − ABA )s = B − ABA, for some/all s > 2. s s h113i r(AB) = r(A) = r(B) and ( A − BAB ) + ( B − ABA ) = A + B − ABA − BAB for some/all s > 2. h114i r(AB) = r(A) = r(B) with one/both of the two facts A − BAB < 0, B − ABA < 0. h115i r(AB) = r(A) = r(B) and A + B − ABA − BAB < 0. m m h116i r(AB) = r(A) = r(B) with one/both of the two facts: A − BAB ∈ COP, B − ABA ∈ COP. m h117i r(AB) = r(A) = r(B) and ( A + B − ABA − BAB )/2 ∈ COP. m m h118i r(AB) = r(A) = r(B) and ABA + ABe Ae ∈ COP, BAB + BAe Be ∈ COP. h119i r(AB) = r(A) = r(B) with one/both of the two facts: ABA + ABe Ae = B, BAB + BAe Be = A. h120i r(AB) = r(A) = r(B) with one/both of the two facts: ABAe + ABAe = 0, BABe + BABe = 0. h121i r(AB) = r(A) = r(B) with one/both of the two facts:(AB)† = BA, (BA)† = AB. h122i r(AB) = r(A) = r(B) with one/both of the two facts:(AB)† = AB, (BA)† = BA. h123i r(AB) = r(A) = r(B) and (AB)† + (BA)† = AB + BA. h124i r(AB) = r(A) = r(B) and [(AB)2 ]† = [ (AB)† ]2. † † m h125i r(AB) = r(A) = r(B) and [(AB) + (BA) ]/2 ∈ COP. h126i r(AB) = r(A) = r(B) with one/both of the two facts: AB(AB)† = ABA, BA(BA)† = BAB. h127i r(AB) = r(A) = r(B) and AB(AB)† + BA(BA)† = ABA + BAB. h128i r(AB) = r(A) = r(B) and (AB)(BA)† = (BA)(AB)†. h129i r(AB) = r(A) = r(B) and [(AB)(AB)†][(BA)(BA)†] = [(BA)(BA)†][(AB)(AB)†]. h130i r(AB) = r(A) = r(B) and A(AeBe)†B = 0, and/or r(AB) = r(A) = r(B) and B(BeAe)†A = 0. h131i r(AB) = r(A) = r(B) and A(AeBe)†B + B(BeAe)†A = 0. # m # m h132i r(AB) = r(A) = r(B) with one/both of the two facts:(AB) ∈ COP, (BA) ∈ COP. # # m h133i r(AB) = r(A) = r(B) and [(AB) + (BA) ]/2 ∈ COP. h134i r(AB) = r(A) = r(B) with one/both of the two facts:(AB)# = (AB)†, (BA)# = (BA)†. h135i r(AB) = r(A) = r(B) and (AB)# + (BA)# = (AB)† + (BA)†. h136i r(AB) = r(A) = r(B) with one/both of the two facts:(AB)(AB)# = (AB)(AB)†, (BA)(BA)# = (BA)(BA)†. h137i r(AB) = r(A) = r(B) and (AB)(AB)# + (BA)(BA)# = (AB)(AB)† + (BA)(BA)†. h138i r(AB) = r(A) = r(B) with one/both of the two facts:(AB)# = AB, (BA)# = BA. h139i r(AB) = r(A) = r(B) and (AB)# + (BA)# = AB + BA. h140i r(AB) = r(A) = r(B) with one/both of the two facts:(ABA)† = ABA, (BAB)† = BAB. h141i r(AB) = r(A) = r(B) and (ABA)† + (BAB)† = ABA + BAB. † † h142i r(AB) = r(A) = r(B) with one/both of the two facts:( Im − AB ) = Im − AB, ( Im − BA ) = Im − BA. Linear subspaces and orthogonal projectors Ë 199

† † h143i r(AB) = r(A) = r(B) with one/both of the two facts:( Im − AB ) = Im − BA, ( Im − BA ) = Im − AB. † † h144i r(AB) = r(A) = r(B) and ( Im − AB ) + ( Im − BA ) = 2Im − AB − BA. 2 † † 2 h145i r(AB) = r(A) = r(B) and [( Im − AB ) ] = [ ( Im − AB ) ] . † † h146i r(AB) = r(A) = r(B) with one/both of the two facts: A( Im − BA ) B = 0, B( Im − AB ) A = 0. † † h147i r(AB) = r(A) = r(B) and A( Im − BA ) B + B( Im − AB ) A = 0. † † m h148i r(AB) = r(A) = r(B) and [( Im − AB ) + ( Im − BA ) ]/2 ∈ COP. † h149i r(AB) = r(A) = r(B) with one/both of the two facts: ( 2Im − A − B )( 2Im − A − B ) = Im − AB, ( 2Im − A − † B )( 2Im − A − B ) = Im − BA. † h150i r(AB) = r(A) = r(B) with one/both of the two facts: ( 2Im − AB − BA )( 2Im − AB − BA ) = Im − AB, † ( 2Im − AB − BA )( 2Im − AB − BA ) = Im − BA. † † h151i r(AB) = r(A) = r(B) with one/both of the two facts:( Im − ABA ) = Im − ABA, ( Im − BAB ) = Im − BAB. † † h152i r(AB) = r(A) = r(B) and ( Im − ABA ) + ( Im − BAB ) = 2Im − ABA − BAB. † † m h153i r(AB) = r(A) = r(B) and [(ABA) + (BAB) ]/2 ∈ COP. † † h154i r(AB) = r(A) = r(B) with one/both of the two facts: A( Im − ABA ) B = 0, A( Im − BAB ) B = 0. † † h155i r(AB) = r(A) = r(B) and A( Im − ABA ) B + A( Im − BAB ) B = 0. † h156i r(AB) = r(A) = r(B) with one/both ( 2Im − ABA − BAB )( 2Im − ABA − BAB ) = Im − ABA, ( 2Im − ABA − † BAB )( 2Im − ABA − BAB ) = Im − BAB. s h157i r(AB) = r(A) = r(B) and Trace[ (AB) ] = Trace(AB) for some/all s > 2. h158i r(AB) = r(A) = r(B) and r(AB) = Trace(AB). s h159i r(AB) = r(A) = r(B) and Trace[ ( Im − AB ) ] = Trace( Im − AB ) for some/all s > 2. s h160i r(AB) = r(A) = r(B) and Trace[ ( Im − ABA ) ] = Trace( Im − ABA ), and/or r(AB) = r(A) = r(B) and s Trace[ ( Im − BAB ) ] = Trace( Im − BAB ) for some/all s > 2. h161i r(AB) = r(A) = r(B) and r[ A, B ] = Trace(A) + Trace(B) − Trace(AB). h162i r(AB) = r(A) = r(B) and r( Im − A − B ) = m − r(A) − r(B) + 2r(AB). h163i r(AB) = r(A) = r(B) and r( Im − A − B ) = m − Trace(A) − Trace(B) + 2Trace(AB). h164i r(AB) = r(A) = r(B) and r( A − B ) + r( Im − A − B ) = m. h165i r(AB) = r(A) = r(B) and 2r[ A, B ] = 2r(A) + 2r(B) − r( AB + BA ). h166i r(AB) = r(A) = r(B) and r( Im − AB ) = m − r(AB). h167i r(AB) = r(A) = r(B) and r( Im − AB ) = m − Trace(AB). h168i r(AB) = r(A) = r(B) with one/both Trace(A) = Trace(AB), Trace(B) = Trace(BA). h169i r(AB) = r(A) = r(B) with one/both r( Im − ABA ) = m − r(ABA), r( Im − BAB ) = m − r(BAB). h170i r(AB) = r(A) = r(B) with one/both r( Im − ABA ) = m − Trace(ABA), r( Im − BAB ) = m − Trace(BAB). h171i r(AB) = r(A) = r(B) with one/both BAB = A, and ABA = B. h172i r(AB) = r(A) = r(B) with one/both A = PM∩N, B = PM∩N. h173i r(AB) = r(A) = r(B) and A + B = 2PM∩N. h174i r(AB) = r(A) = r(B) and Im − A − B = −PM∩N + PM⊥∩N⊥ . ⊥ ⊥ h175i r(AB) = r(A) = r(B) and R( Im − A − B ) = (M ∩ N) ⊕ (M ∩ N ). h i ⊥ ⊥ 176 r(AB) = r(A) = r(B) and PR( Im−A−B ) = PM∩N + PM ∩N . m h177i r(AB) = r(A) = r(B) and R(A − B) ⊕ R(Im − A − B) = C . h178i r(AB) = r(A) = r(B) with one/both AB = PM∩N, BA = PM∩N. h179i r(AB) = r(A) = r(B) and AB + BA = 2PM∩N. h180i r(AB) = r(A) = r(B) with one/both R(AB) = M ∩ N, R(BA) = M ∩ N. h181i r(AB) = r(A) = r(B) with one/both AB = PR(AB), BA = PR(BA). h182i r(AB) = r(A) = r(B) and AB + BA = PR(AB) + PR(BA). ⊥ ⊥ h183i r(AB) = r(A) = r(B) and R( Im − AB ) = R( Im − BA ) = M ∩ N . h184i r(AB) = r(A) = r(B) and Im − AB = PM⊥∩N⊥ . h185i r(AB) = r(A) = r(B) with one/both R(AB) ∩ R( Im − AB ) = {0}, R(BA) ∩ R( Im − BA ) = {0}. m m h186i r(AB) = r(A) = r(B) with one/both C = R(AB) ⊕ R( Im − AB ), C = R(BA) ⊕ R( Im − BA ). h i 187 r(AB) = r(A) = r(B) with one/both Im = PR(AB) + PR( Im−AB ), Im = PR(BA) + PR( Im−BA ). h188i r(AB) = r(A) = r(B) with one/both ABA = PM∩N, BAB = PM∩N. h189i r(AB) = r(A) = r(B) and ABA + BAB = 2PM∩N. 200 Ë Yongge Tian h190i r(AB) = r(A) = r(B) and R( AB + BA ) = M ∩ N. h191i r(AB) = r(A) = r(B) with one/both R(AB) = R(BA) N (AB) = N (BA). m m h192i r(AB) = r(A) = r(B) with one/both R(AB) ⊕ N (AB) = C , R(BA) ⊕ N (BA) = C . h193i r(AB) = r(A) = r(B) with one/both R(AB) = R(A) ∩ R(B), R(BA) = R(A) ∩ R(B). h194i r(AB) = r(A) = r(B) with one/both R(AeBe) = N (A) ∩ N (B), R(BeAe) = N (A) ∩ N (B). h195i r(AB) = r(A) = r(B) with one/both R(ABAe) = R(BABe), N (ABAe) = N (BABe).

7 Equalities and inequalities for norms of orthogonal projectors

Norms of orthogonal projectors and their operations are an important subject in both matrix theory and applications. Many equalities and inequalities for the spectrum norm and F-norms of orthogonal projectors and 2 ∗ their operations can be found in the literature; see, e.g., [42, 71, 86]. Applying the F-norm kMkF = Trace(MM ) to parts of the matrix equalities in Theorem 3.4(a) and simplifying by (3.7)–(3.9), we obtain the following norm equalities and inequalities.

m Theorem 7.1. Let M and N be two linear subspaces of C , and assume that A, B, Ae, and Be in (3.2) are decomposed as (3.3)–(3.6), respectively. Then, the following norm identities hold

2 2 kA + BkF = 2Trace(C ) + 2k1 + 4k3 + k4 + k5, 2 2 kA + B − ImkF = 2Trace(C ) + k3 + k6, 2 2 kA + B − 2PM∩N − PM∩N⊥ − PM⊥∩NkF = 2Trace(C ) + 2k1, 2 2 kP(M+N) − A − BkF = 2Trace(C ) + k1, 2 2 kP(M+N) − A − B + ABkF = Trace(C ), 2 2 kP(M+N) − A − B + PM∩NkF = 2Trace(C ), 2 2 kA − BkF = 2Trace(S ) + k4 + k5, 2 2 kA − B − PM∩N⊥ + PM⊥∩NkF = 2Trace(S ), 2 2 2 kABkF = kBAkF = Trace(C ) + k3, 2 2 kAB − PM∩NkF = Trace(C ), 2 2 4 kAB + BAkF = 2Trace( C + C ) + 4k3, 2 2 4 kAB + BA − 2PM∩NkF = 2Trace( C + C ), 2 2 2 2 2 2 2 kAB − BAkF = kABAe + ABAe kF = kBABe + BABe kF = 2kABAekF = 2kBABekF = 2Trace(C S ), 2 2 kIm − ABkF = Trace(S ) + k1 + k4 + k5 + k6, 2 2 2 2 2 2 kABekF = Trace(S ) + k4, kABe kF = Trace(S ) + k5, kAeBekF = Trace(C ) + k6, 2 2 4 kABAkF = kBABkF = Trace(C ) + k3, 2 2 4 kABA − PM∩NkF = kBAB − PM∩NkF = Trace(C ), 2 2 4 kABAe kF = kBAe BekF = Trace(S ) + k4, 2 2 4 kBABe kF = kABe AekF = Trace(S ) + k5, 2 2 4 kAeBeAekF = kBeAeBekF = Trace(C ) + k6, 2 6 4 2 4 2 kABA + BABkF = 2Trace( C + C ) + 4k3, kABA − BABkF = 2Trace(C S ), 2 6 4 kABA + BAB − 2PM∩NkF = 2Trace( C + C ), 2 2 4 kIm − ABAkF = kIm − BABkF = Trace(S ) + k1 + k4 + k5 + k6, 2 4 2 4 kA − BABkF = −Trace(C ) + k1 + k4, kB − ABAkF = −Trace(C ) + k1 + k5, 2 4 4 kABA + ABe AekF = Trace(C + S ) + k3 + k5, Linear subspaces and orthogonal projectors Ë 201 and

† 2 −4 −4 2 −1 k(A + B) kF = 2Trace(2S + S C ) + 4 k3 + k4 + k5, † 2 −2 2 k(A + B − Im) kF = 2Trace(C S ) + 2k1 + k3 + k6, † 2 −2 k(A − B) kF = 2Trace(S ) + k4 + k5, † 2 † 2 −2 k(AB) kF = k(BA) kF = Trace(C ) + k3, † 2 −4 −2 −4 −1 k(AB + BA) kF = 2Trace( S + C S ) + 4 k3, † 2 † 2 † 2 † 2 k(AB − BA) kF = k(ABAe + ABAe ) kF = k(BABe + BABe ) kF = 2k(ABAe) kF † 2 −2 −2 = 2k(BABe) kF = 2Trace(C S ), † 2 −2 −4 k(Im − AB) kF = Trace(S + S ) + k1 + k4 + k5 + k6, † 2 −2 † 2 −2 † 2 −2 k(ABe) kF = Trace(S ) + k4, k(ABe ) kF = Trace(S ) + k5, k(AeBe) kF = Trace(C ) + k6, † 2 † 2 −4 k(ABA) kF = k(BAB) kF = Trace(C ) + k3, † 2 † 2 −4 k(ABAe ) kF = k(BAe Be) kF = Trace(S ) + k4, † 2 † 2 −4 k(BABe ) kF = k(ABe Ae) kF = Trace(S ) + k5, † 2 † 2 −4 k(AeBeAe) kF = k(BeAeBe) kF = Trace(C ) + k6, † 2 −2 −4 −4 −4 −1 k(ABA + BAB) kF = 2Trace( C S + C S ) + 4 k3, † 2 −4 −2 k(ABA − BAB) kF = 2Trace(C S ), † 2 † 2 −4 k(Im − ABA) kF = k(Im − BAB) kF = Trace(S ) + k1 + k4 + k5 + k6, † 2 −4 −2 −2 −2 k(A − BAB) kF = Trace(C S + C S ) + k4, † 2 −4 −2 −2 −2 k(B − ABA) kF = Trace(C S + C S ) + k5, † 2 −4 −4 k(ABA + ABe Ae) kF = Trace(C + S ) + k3 + k5.

Substituting (3.8) and (3.9) into the rst group of equalities in Theorem 7.1 and simplifying, we obtain the following norm inequalities.

m Corollary 7.2. Let M and N be two linear subspaces of C , and A, B, Ae, and Be be as given in (3.2). Then, the following norm inequalities hold

2 3r(A) + 3r(B) − 2r[ A, B ] 6 kA + BkF 6 r(A) + r(B) + 2r(AB), 2 m + r(A) + r(B) − 2r[ A, B ] 6 kA + B − ImkF 6 m − r(A) − r(B) + 2r(AB), 2 r(A) + r(B) − 2r(AB) 6 kA − BkF 6 2r[ A, B ] − r(A) − r(B), 2 2 r(A) + r(B) − r[ A, B ] 6 kABkF = kBAkF 6 r(AB), 2 4r(A) + 4r(B) − 4r[ A, B ] 6 kAB + BAkF 6 4r(AB), 1 1 1 1 kAB − BAk2 r[ A, B ] − r(A) − r(B) + r(AB), F 6 2 2 2 2 2 m − r(AB) 6 kIm − ABkF 6 m − r(A) − r(B) + r[ A, B ], 2 r(A) − r(AB) 6 kABekF 6 r[ A, B ] − r(B), 2 r(B) − r(AB) 6 kABe kF 6 r[ A, B ] − r(A), 2 m − r[ A, B ] 6 kAeBekF 6 m − r(A) − r(B) + r(AB), 2 2 r(A) + r(B) − r[ A, B ] 6 kABAkF = kBABkF 6 r(AB), 2 2 r(A) − r(AB) 6 kABAe kF = kBAe BekF 6 r[ A, B ] − r(B), 2 2 r(B) − r(AB) 6 kBABe kF = kABe AekF 6 r[ A, B ] − r(A), 2 2 m − r[ A, B ] 6 kAeBeAekF = kBeAeBekF 6 m − r(A) − r(B) + r(AB), 202 Ë Yongge Tian

2 2r(A) + 2r(B) − 2r[ A, B ] 6 kABA + BABkF 6 2r(AB), 8 8 8 8 kABA − BABk2 r[ A, B ] − r(A) − r(B) + r(AB), F 6 27 27 27 27 2 2 m − r(AB) 6 kIm − ABAkF = kIm − BABkF 6 m − r(A) − r(B) + r[ A, B ], 2 r(A) − r(AB) 6 kA − BABkF 6 r[ A, B ] − r(B), 2 r(B) − r(AB) 6 kB − ABAkF 6 r[ A, B ] − r(A), 1 3 1 1 2 r(A) + r(B) − r(AB) − r[ A, B ] kABA + ABe Aek r(B). 2 2 2 2 6 F 6 In particular, if r[ A, B ] = r(A) + r(B) − r(AB), then

2 2 kA + BkF = r(A) + r(B) + 2r(AB), kA + B − ImkF = m − r(A) − r(B) + 2r(AB), 2 2 2 kA − BkF = r(A) + r(B) − 2r(AB), kABkF = kBAkF = r(AB), 2 2 kAB + BAkF = 4r(AB), kIm − ABkF = m − r(AB), 2 2 kABekF = r(A) − r(AB), kABe kF = r(B) − r(AB), 2 2 2 kAeBekF = m − r(A) − r(B) + r(AB), kABAkF = kBABkF = r(AB), 2 2 2 2 kABAe kF = kBAe BekF = r(A) − r(AB), kBABe kF = kABe AekF = r(B) − r(AB), 2 2 2 kAeBeAekF = kBeAeBekF = m − r(A) − r(B) + r(AB), kABA + BABkF = 2r(AB), 2 2 2 kIm − ABAkF = kIm − BABkF = m − r(AB), kA − BABkF = r(A) − r(AB), 2 2 kB − ABAkF = r(B) − r(AB), kABA + ABe AekF = r(B).

8 Characterization of an EP matrix

The huge amount of formulas, facts, and results in the previous sections are milestones in linear algebra and matrix theory, which can serve as building materials and fundamental tools in many elds of mathematics and other disciplines. As a direct application, we consider a well-known problem of characterizing commutativity of a square matrix and its Moore–Penrose inverse. m×m A matrix M ∈ C is said to be EP (or range Hermitian) if

∗ R(M ) = R(M) holds. This concept was introduced by Schwerdtfeger in [127] and has been studied in detail by lots of authors; see e.g., [13, 23, 39, 40, 58–62, 67, 84, 85, 88, 90, 91, 104, 128, 144, 152] and references therein. For convenience of representation, we denote by

† † † † A = MM , B = M M, Ae = Im − MM , Be = Im − M M the four orthogonal projectors induced by M. Then it is obvious that

2 r(A) = r(B) = r(M), r(Ae) = r(Be) = m − r(M), r(AB) = r(BA) = r(M ).

Many identifying conditions have been established for a square matrix to be EP in the literature. As a summary, we collect various known results (see [144]) on EP matrix, and present below many novel equivalent statements for a matrix to be EP without proofs.

m×m Theorem 8.1. Let M ∈ C be given, and k1, ... , k6 be as given in (3.15) and (3.16). Then, the following statements are equivalent: h1i M is EP and/or M∗ is EP. h2i R⊥(M∗) = R⊥(M). h3i Some/all of the four facts: R(M) ⊆ R(M∗), R(M) ⊇ R(M∗), R⊥(M) ⊆ R⊥(M∗), R⊥(M) ⊇ R⊥(M∗). h4i R(M) + R(M∗) = R(M) and/or R(M) + R(M∗) = R(M∗). h5i R⊥(M) + R⊥(M∗) = R⊥(M) and/or R⊥(M) + R⊥(M∗) = R⊥(M∗). Linear subspaces and orthogonal projectors Ë 203 h6i R(M) + R(M∗) = R(M) ∩ R(M∗) and/or R⊥(M) + R⊥(M∗) = R⊥(M) ∩ R⊥(M∗). h7i Some/all of the four facts: R(M) = R(M) ∩ R(M∗), R(M∗) = R(M) ∩ R(M∗), R⊥(M) = R⊥(M) ∩ R⊥(M∗), R⊥(M∗) = R⊥(M) ∩ R⊥(M∗). h8i R(M) = R(M∗) ⊕ (R(M) ∩ R⊥(M∗)) and R(M∗) = R(M) ⊕ (R⊥(M) ∩ R(M∗)). h9i R(M) = R(M∗) ∩ (R(M) + R⊥(M∗)) and R(M∗) = R(M) ∩ (R⊥(M) + R(M∗)). h10i R⊥(M) = R⊥(M∗) ⊕ (R⊥(M) ∩ R(M∗)) and R⊥(M∗) = R⊥(M) ⊕ (R(M) ∩ R⊥(M∗)). h11i R⊥(M) = R⊥(M∗) ∩ (R⊥(M) + R(M∗)) and R⊥(M∗) = R⊥(M) ∩ (R(M) + R⊥(M∗)). ∗ ⊥ ⊥ ∗ m ∗ ⊥ ⊥ ∗ h12i (R(M) ∩ R(M )) ⊕ (R (M) ∩ R (M )) = C and/or (R(M) + R(M )) ∩ (R (M) + R (M )) = {0}. m ⊥ ∗ m ∗ ⊥ h13i C = R(M) ⊕ R (M ) and/or C = R(M ) ⊕ R (M). h14i 2 dim( R(M) + R(M∗) ) = dim(R(M)) + dim(R(M∗)) and/or 2 dim( R⊥(M) + R⊥(M∗) ) = dim(R⊥(M)) + dim(R⊥(M∗)). h15i dim( R(M) + R(M∗) ) = dim(R(M)) = dim(R(M∗)) and/or dim( R⊥(M) + R⊥(M∗) ) = dim(R⊥(M)) = dim(R⊥(M∗)). h16i dim(R(M) + R(M∗)) = dim(R(M) ∩ R(M∗)) and/or dim( R⊥(M) + R⊥(M∗) ) = dim(R⊥(M) ∩ R⊥(M∗)). h17i dim(R(M) ∩ R(M∗)) = dim(R(M)) = dim(R(M∗)) and/or dim(R⊥(M) ∩ R⊥(M∗)) = dim(R⊥(M)) = dim(R⊥(M∗)). h18i dim(R(M) ∩ R(M∗)) + dim(R⊥(M) ∩ R⊥(M∗)) = m. h19i dim(R(M) + R(M∗)) + dim(R⊥(M) + R⊥(M∗)) = m. h20i R(M) ∩ R⊥(M∗) = R⊥(M) ∩ R(M∗) = {0} with some/all of the four facts: R(M) and R(M∗) are commutative, R(M) and R⊥(M∗) are commutative, R⊥(M) and R(M∗) are commutative, R⊥(M) and R⊥(M∗) are commutative. h21i R(M) ∩ R⊥(M∗) = R⊥(M) ∩ R(M∗) = {0} with some/all of the four facts: R(M) ◦ R⊥(M∗) = {0}, R(M∗) ◦ R⊥(M) = {0}, R⊥(M) ◦ R(M∗) = {0}, R⊥(M∗) ◦ R(M) = {0}. ⊥ ∗ ⊥ ∗ ⊥ ⊥ ∗ m h22i R(M)∩R (M ) = R (M)∩R(M ) = {0} with some/all of the four facts: R(M)⊕(R (M)∩R (M )) = C , ∗ ⊥ ⊥ ∗ m ⊥ ∗ m ⊥ ∗ ∗ m R(M ) ⊕ (R (M) ∩ R (M )) = C , R (M) ⊕ (R(M) ∩ R(M )) = C , R (M ) ⊕ (R(M) ∩ R(M )) = C . h23i R(M)∩R⊥(M∗) = R⊥(M)∩R(M∗) = {0} with some/all of the four facts: R(M)◦R(M∗) = R(M∗)◦R(M), R(M) ◦ R⊥(M∗) = R⊥(M∗) ◦ R(M), R⊥(M) ◦ R(M∗) = R(M∗) ◦ R⊥(M), R⊥(M) ◦ R⊥(M∗) = R⊥(M∗) ◦ R⊥(M). h24i R(M) ∩ R⊥(M∗) = R⊥(M) ∩ R(M∗) = {0} with some/all of the four facts: R(M) ⊕ (R⊥(M) ∩ R⊥(M∗)) = R⊥(M∗) ⊕ (R(M) ∩ R(M∗)), R⊥(M) ⊕ (R(M) ∩ R(M∗)) = R(M∗) ⊕ (R⊥(M) ∩ R⊥(M∗)). h25i R(M) ∩ R⊥(M∗) = R⊥(M) ∩ R(M∗) = {0} with some/all of the four facts: R(M) • R(M∗) = {0}, R(M∗) • R(M) = {0}, R⊥(M) • R(M∗) = {0}, R⊥(M∗) • R(M) = {0}. h26i M is EP, i.e., R(M) = R(MT). h27i MT is EP. h28i MM∗M is EP. h29i M† is EP. h30i r(M) = r(M2) and M# is EP. h31i UMU∗ is EP for any/some unitary matrix U. h32i PMP∗ is EP for any/some nonsingular matrix P. " # M 0 h33i There exists a unitary matrix U such that UMU∗ = 1 , where M is nonsingular. 0 0 1 " # M 0 h34i There exists a nonsingular matrix P such that PMP∗ = 1 , where M is nonsingular. 0 0 1 " ∗ # ∗ M1 M1Z h35i M can be represented as PMP = ∗ , where P is a permutation matrix and M1 is ZM1 ZM1Z nonsingular. ∗ ∗ ∗ ∗ h36i There exists a matrix V such that M = MV and/or M = V1M, M = M V2, M = V3M . h37i There exists a matrix V such that M† = MV and/or M† = VM, M = M†V, M = VM†. h38i r(M) = r(M2) and there exists a matrix V such that M# = M∗V and/or M# = VM∗. h39i r(M) = r(M2) and M† = M#. 204 Ë Yongge Tian h40i M commutes with MM† and/or M commutes with M†M. h41i M† commutes with MM† and/or M† commutes with M†M. k k h42i r(M) = r(M ) and M is EP for any/some positive number k > 2. h43i A = B and/or Ae = Be. h44i Some/all of the four facts: AB = B, BA = A, AeBe = Be, BeAe = Ae. h45i ABA = A and BAB = B. h46i ABA = B and BAB = A. h47i R(M) = R(A − B) ⊕ R(M∗) and R(M∗) = R(A − B) ⊕ R(M). h48i R⊥(M) = R(A − B) ⊕ R⊥(M∗) and R⊥(M∗) = R⊥(M) ⊕ R(A − B). h49i 2r[ M, M∗ ] = r(M) + r(M∗), i.e., 2r[ A, B ] = r(A) + r(B). h50i r[ M, M∗ ] = r(M) = r(M∗), i.e., r[ A, B ] = r(A) = r(B). h51i r[ M, M∗ ] = r(M2), i.e., r[ A, B ] = r(AB). h52i 2r[ Ae, Be ] = r(Ae) + r(Be). h53i r[ Ae, Be ] = r(Ae) = r(Be). h54i r[ Ae, Be ] = r(AeBe). h55i There exists a unitary matrix U such that

∗ αA + βB = Udiag (α + β)Ik3 , 0k6 U

for some/all two nonzero real numbers α and β. h56i There exists a unitary matrix U such that

∗ αAB + βBA = Udiag (α + β)Ik3 , 0m−k3 U

for some/all two nonzero real numbers α and β. h57i There exists a unitary matrix U such that

∗ αABA + βBAB = Udiag (α + β)Ik3 , 0m−k3 U

for some/all two nonzero real numbers α and β. h58i There exists a unitary matrix U such that

∗ αA + βBAB = Udiag (α + β)Ik3 , 0m−k3 U ,

for some/all two nonzero real numbers α and β, and/or R(M) ∩ R⊥(M∗) = R⊥(M) ∩ R(M∗) = {0} and there exists a unitary matrix U such that

∗ αB + βABA = Udiag (α + β)Ik3 , 0m−k3 U

for some/all two nonzero real numbers α and β. h59i There exists a unitary matrix U such that

† † ∗ α(AB) + β(BA) = Udiag (α + β)Ik3 , 0m−k3 U

for some/all two nonzero real numbers α and β. h60i There exists a unitary matrix U such that

† † ∗ α(ABA) + β(BAB) = Udiag (α + β)Ik3 , 0m−k3 U

for some/all two nonzero real numbers α and β. h61i r(M) = r(M2) and (M2)† = (M†)2. h62i r(M) = r(M2) and M2(M†)2M2 = M2. h63i r(M) = r(M2) and M(M†)2M = MM#. h64i There exists a polynomial p(x) such that M† = p(M) and/or there exists a polynomial q(x) such that M = q(M†). Linear subspaces and orthogonal projectors Ë 205 h65i (MM†)2 = M2(M†)2 and/or (M†M)2 = (M†)2M2. h66i r(M) = r(M2) and (MM†)(M†M) = (M†M)(MM†). h67i r(M) = r(M2) and (MM†)(M∗M) = (M∗M)(MM†) and/or (M†M)(MM∗) = (MM∗)(M†M). h68i r(M) = r(M2) with one/both of the two facts: MM† commutes with MM∗ + λM∗M for some/all complex numbers λ ≠ 0, M†M commutes with MM∗ + λM∗M for some/all complex numbers λ ≠ 0. h69i M2M† + M†M2 = 2M. h70i M∗M#M + MM#M∗ = 2M∗. h71i M†M#M + MM#M† = 2M†. h72i M2M† + (M2M†)∗ = M + M∗ and/or M†M2 + (M†M2)∗ = M + M∗. " # ∗ M11 M12 h73i For any unitary matrix U for which UMU = with M11 and M22 square, M11 and M22 are 0 M22 ∗ ∗ EP, R(M12) ⊆ R(M11), and R(M ) ⊆ R(M ). 12 22 " # ∗ M11 M12 h74i For any nonsingular matrix P for which PMP = with M11 and M22 square, M11 and M22 0 M22 ∗ ∗ are EP, R(M12) ⊆ R(M11), and R(M12) ⊆ R(M22). s+t s+t † s † t † h75i r(M) = r(M ) and (M ) = (M ) (M ) for some/all integers s, t > 1. h76i M†N = NM† whenever MN = NM for certain matrix N. h77i MM†( M+λM† ) = ( M+λM† )MM† and/or M†M( M+λM† ) = ( M+λM† )M†M for some/all complex numbers λ ≠ 0. h78i MM†( M + λM∗ ) = ( M + λM∗ )MM† and/or M†M( M + λM∗ ) = ( M + λM∗ )M†M for some/all complex number λ ≠ 0. h79i R( M + λM† ) = R( λM + M3 ) for some/all complex number λ ≠ 0. h80i N ( M + λM† ) = N ( λM + M3 ) for some/all complex number λ ≠ 0. h81i r(M) = r(M2) and MM# is Hermitian. h82i r(M) = r(M2) with one/both of the two facts: MM† = MM#, M†M = M#M. h83i r(M) = r(M2) and MM#M∗ = M∗M#M. h84i r(M) = r(M2) and MM#M† = M†M#M. h85i r(M) = r(M2) with one/both of the two facts:(MM∗)(MM#) = (MM#)(MM∗), (M∗M)(M#M) = (M#M)(M∗M). h86i r(M) = r(M2) and (MM#)(MM∗ + λM∗M) = (MM∗ + λM∗M)(MM#) for some/all complex numbers λ ≠ 0. h87i {r(M) = r(M2) with one/both of the two facts:(MM†)(M∗M)† = (M∗M)†(MM†), (M†M)(MM∗)† = (MM∗)†(M†M). k k † † k h88i r(M) = r(M ) and M M = M M for some/all integers k > 2. k k+1 † † k+1 k h89i r(M) = r(M ) and M M + M M = 2M for some/all integers k > 1. k k+1 † † k+1 ∗ k k ∗ h90i r(M) = r(M ) and M M + (M M ) = M + (M ) for some/all integers k > 1. h91i r(M2) = r(M) with some/all of the four facts: AB = BA, ABe = BAe , ABe = BAe, AeBe = BeAe. h92i r(M2) = r(M) with some/all of the four facts: ABA = BAB, ABAe = BAe Be,, ABe Ae = BABe , AeBeAe = BeAeBe. 2 h93i r(M ) = r(M) with one/both of the two facts: AB + BA 4 A, AB + BA 4 B. 2 h94i r(M ) = r(M) and AB + BA < 0. 2 † † m h95i r(M ) = r(M) and A( A + B ) A + B( A + B ) B ∈ COP. h96i r(M2) = r(M) with one/both of the two facts: A( A + B )† = ( A + B )†A, B( A + B )† = ( A + B )†B. 2 2 2 h97i r(M ) = r(M) with one/both of the two facts:( Im − A − B ) < Im − A − B, ( A + B − Im ) < A + B − Im . 2 2 m h98i r(M ) = r(M) and ( I − A − B ) ∈ COP. 2 2s+1 h99i r(M ) = r(M) and ( Im − A − B ) = Im − A − B for one/both of s > 1. 2 † h100i r(M ) = r(M) and ( Im − A − B ) = Im − A − B. h101i r(M2) = r(M) with one/both of the two facts:( A + B )( A + B )† = A + B − AB, ( A + B )( A + B )† = A + B − BA. h102i r(M2) = r(M) and [( A + B )( A + B )† − A ][ ( A + B )( A + B )† − B ] = 0. h103i r(M2) = r(M) with one/both of the two facts:( A + B )( A + B )† = A + ABe Ae, ( A + B )( A + B )† = B + BAe Be. h104i r(M2) = r(M) with some/all of the four facts:( A+B )( A+B )† = (AB)(AB)†, ( A+B )( A+B )† = (AB)†(AB), ( A + B )( A + B )† = (BA)(BA)†, ( A + B )( A + B )† = (BA)†(BA). 206 Ë Yongge Tian

2 † † † h105i r(M ) = r(M) with some/all of the four facts: A < (BA)(BA) , A < (AB) (AB), B < (AB)(AB) , B < (BA)†(BA). 2 † † † h106i r(M ) = r(M) with one/both of the two facts: A + B < (AB)(AB) + (BA)(BA) , A + B < (AB) (AB) + (BA)†(BA). 2 † † † h107i r(M ) = r(M) with some/all of the four facts: A < (BAe )(BAe ) , A < (ABe) (ABe), B < (ABe )(ABe ) , B < (BAe)†(BAe). 2 † † † h108i r(M ) = r(M) with one/both of the two facts: A + B < (ABe )(ABe ) + (BAe )(BAe ) , A + B < (BAe) (BAe) + (ABe)†(ABe). h109i r(M2) = r(M) with one/both of the two facts: 2A( AB + BA )†B = AB, 2B( AB + BA )†A = BA. h110i r(M2) = r(M) and 2A( AB + BA )†B + 2B( AB + BA )†A = AB + BA. 2 s m h111i r(M ) = r(M) and (AB) ∈ COP for one/both of s > 1. 2 s h112i r(M ) = r(M) and (AB) = AB for one/both of s > 2. 2 s s h113i r(M ) = r(M) and (AB) = (BA) for some/all of s > 1. 2 s h114i r(M ) = r(M) and (AB) = BA for some/all of s > 2. 2 s s h115i r(M ) = r(M) and (AB) + (BA) = AB + BA for some/all of s > 2. 2 s m s m h116i r(M ) = r(M) with one/both of the two facts:(ABA) ∈ COP, (BAB) ∈ COP for some/all of s > 1. 2 s s h117i r(M ) = r(M) with one/both of the two facts:(ABA) = ABA, (BAB) = BAB for some/all s > 2. 2 s s h118i r(M ) = r(M) with one/both of the two facts:(ABA) = BAB, (BAB) = ABA for some/all s > 1. 2 s s h119i r(M ) = r(M) and (ABA) + (BAB) = ABA + BAB for some/all of s > 2. 2 s s h120i r(M ) = r(M) with one/both of the two facts:(ABA) = AB, (BAB) = BA for some/all of s > 1. 2 s s h121i r(M ) = r(M) and (ABA) + (BAB) = AB + BA for some/all of s > 1. 2 s h122i r(M ) = r(M) and ( Im − AB ) = Im − AB for some/all of s > 2. 2 s h123i r(M ) = r(M) and ( Im − AB ) = Im − BA for some/all of s > 2. 2 s s h124i r(M ) = r(M) and ( Im − AB ) = ( Im − BA ) for some/all of s > 1. 2 s h125i r(M ) = r(M) and ( A + B − AB ) = A + B − AB for some/all of s > 2. h126i r(M2) = r(M) and ( A + B − AB )† = A + B − AB. 2 m h127i r(M ) = r(M) and A + B − AB ∈ COP. 2 2s 2s−1 h128i r(M ) = r(M) and ( AB + BA ) = 2 ( AB + BA ) for some/all of s > 1. 2 2 h129i r(M ) = r(M) and ( Im − AB − BA ) = Im . 2 − − h130i r(M ) = r(M) with one/both the two facts: Im + AB ∈ {( Im − AB ) }, Im + BA ∈ {( Im − BA ) }. 2 s s h131i r(M ) = r(M) with one/both the two facts:( Im −ABA ) = Im −ABA, ( Im −BAB ) = Im −BAB for some/all of s > 2. 2 s s h132i r(M ) = r(M) and ( Im − ABA ) + ( Im − BAB ) = 2Im − ABA − BAB for some/all of s > 2. 2 s s−1 h133i r(M ) = r(M) and ( ABA + BAB ) = 2 ( ABA + BAB ) for some/all of s > 2. 2 2 h134i r(M ) = r(M) and ( Im − ABA − BAB ) = Im . h135i r(M2) = r(M) with one/both the two facts:( A − BAB )s = A − BAB, ( B − ABA )s = B − ABA for some/all of s > 2. 2 s s h136i r(M ) = r(M) and ( A − BAB ) + ( B − ABA ) = A + B − ABA − BAB for some/all of s > 2. 2 h137i r(M ) = r(M) with one/both of the two facts: A − BAB < 0, B − ABA < 0. 2 h138i r(M ) = r(M) and A + B − ABA − BAB < 0. 2 m m h139i r(M ) = r(M) with one/both of the two facts: A − BAB ∈ COP, B − ABA ∈ COP. 2 m h140i r(M ) = r(M) and ( A + B − ABA − BAB )/2 ∈ COP. 2 m m h141i r(M ) = r(M) with one/both of the two facts: ABA + ABe Ae ∈ COP, BAB + BAe Be ∈ COP. h142i r(M2) = r(M) with one/both of the two facts: ABA + ABe Ae = B, BAB + BAe Be = A. h143i r(M2) = r(M) with one/both of the two facts: ABAe + ABAe = 0, BABe + BABe = 0. h144i r(M2) = r(M) with one/both of the two facts:(AB)† = BA, (BA)† = AB. h145i r(M2) = r(M) with one/both of the two facts:(AB)† = AB, (BA)† = BA. h146i r(M2) = r(M) and (AB)† + (BA)† = AB + BA. h147i r(M2) = r(M) and [(AB)2 ]† = [ (AB)† ]2. 2 † † m h148i r(M ) = r(M) and [(AB) + (BA) ]/2 ∈ COP. h149i r(M2) = r(M) with one/both of the two facts: AB(AB)† = ABA, BA(BA)† = BAB. Linear subspaces and orthogonal projectors Ë 207 h150i r(M2) = r(M) and AB(AB)† + BA(BA)† = ABA + BAB. h151i r(M2) = r(M) and (AB)(BA)† = (BA)(AB)†. h152i r(M2) = r(M) and [(AB)(AB)†][(BA)(BA)†] = [(BA)(BA)†][(AB)(AB)†]. h153i r(M2) = r(M) with one/both of the two facts: A(AeBe)†B = 0, B(BeAe)†A = 0. h154i r(M2) = r(M) and A(AeBe)†B + B(BeAe)†A = 0. 2 # m # m h155i r(M ) = r(M) with one/both of the two facts:(AB) ∈ COP, (BA) ∈ COP. 2 # # m h156i r(M ) = r(M) and [(AB) + (BA) ]/2 ∈ COP. h157i r(M2) = r(M) with one/both of the two facts:(AB)# = (AB)†, (BA)# = (BA)†. h158i r(M2) = r(M) and (AB)# + (BA)# = (AB)† + (BA)†. h159i r(M2) = r(M) with one/both of the two facts:(AB)(AB)# = (AB)(AB)†, (BA)(BA)# = (BA)(BA)†. h160i r(M2) = r(M) and (AB)(AB)# + (BA)(BA)# = (AB)(AB)† + (BA)(BA)†. h161i r(M2) = r(M) with one/both of the two facts:(AB)# = AB, (BA)# = BA. h162i r(M2) = r(M) and (AB)# + (BA)# = AB + BA. h163i r(M2) = r(M) and one/both of the two facts:(ABA)† = ABA, (BAB)† = BAB. h164i r(M2) = r(M) and (ABA)† + (BAB)† = ABA + BAB. 2 † † h165i r(M ) = r(M) with one/both of the two facts:( Im − AB ) = Im − AB, ( Im − BA ) = Im − BA. 2 † † h166i r(M ) = r(M) with one/both of the two facts:( Im − AB ) = Im − BA, ( Im − BA ) = Im − AB. 2 † † h167i r(M ) = r(M) and ( Im − AB ) + ( Im − BA ) = 2Im − AB − BA. 2 2 † † 2 h168i r(M ) = r(M) and [( Im − AB ) ] = [ ( Im − AB ) ] . 2 † † h169i r(M ) = r(M) with one/both of the two facts: A( Im − BA ) B = 0, B( Im − AB ) A = 0. 2 † † h170i r(M ) = r(M) and A( Im − BA ) B + B( Im − AB ) A = 0. 2 † † m h171i r(M ) = r(M) and [( Im − AB ) + ( Im − BA ) ]/2 ∈ COP. 2 † h172i r(M ) = r(M) with one/both of the two facts: ( 2Im − A − B )( 2Im − A − B ) = Im − AB, ( 2Im − A − B )( 2Im − † A − B ) = Im − BA. 2 † h173i r(M ) = r(M) with one/both of the two facts: ( 2Im − AB − BA )( 2Im − AB − BA ) = Im − AB, ( 2Im − AB − † BA )( 2Im − AB − BA ) = Im − BA. 2 † † h174i r(M ) = r(M) with one/both of the two facts:( Im − ABA ) = Im − ABA, ( Im − BAB ) = Im − BAB. 2 † † h175i r(M ) = r(M) and ( Im − ABA ) + ( Im − BAB ) = 2Im − ABA − BAB. 2 † † m h176i r(M ) = r(M) and [(ABA) + (BAB) ]/2 ∈ COP. 2 † † h177i r(M ) = r(M) with one/both of the two facts: A( Im − ABA ) B = 0, A( Im − BAB ) B = 0. 2 † † h178i r(M ) = r(M) and A( Im − ABA ) B + A( Im − BAB ) B = 0. 2 † h179i r(M ) = r(M) with one/both of the two facts: ( 2Im − ABA − BAB )( 2Im − ABA − BAB ) = Im − ABA, † ( 2Im − ABA − BAB )( 2Im − ABA − BAB ) = Im − BAB. 2 s h180i r(M ) = r(M) and Trace[ (AB) ] = Trace(AB) for some/all of s > 2. h181i r(M2) = r(M) and r(AB) = Trace(AB). 2 s h182i r(M ) = r(M) and Trace[ ( Im − AB ) ] = Trace( Im − AB ) for some/all of s > 2. 2 s h183i r(M ) = r(M) with one/both of the two facts: Trace[ ( Im − ABA ) ] = Trace( Im − ABA ), Trace[ ( Im − s BAB ) ] = Trace( Im − BAB ) for some/all of s > 2. h184i r(M2) = r(M) and r[ A, B ] = Trace(A) + Trace(B) − Trace(AB). 2 h185i r(M ) = r(M) and r( Im − A − B ) = m − r(A) − r(B) + 2r(AB). 2 h186i r(M ) = r(M) and r( Im − A − B ) = m − Trace(A) − Trace(B) + 2Trace(AB). 2 h187i r(M ) = r(M) and r( A − B ) + r( Im − A − B ) = m. h188i r(M2) = r(M) and 2r[ A, B ] = 2r(A) + 2r(B) − r( AB + BA ). 2 h189i r(M ) = r(M) and r( Im − AB ) = m − r(AB). 2 h190i r(M ) = r(M) and r( Im − AB ) = m − Trace(AB). h191i r(M2) = r(M) with one/both of the two facts: Trace(A) = Trace(AB), Trace(B) = Trace(BA). 2 h192i r(M ) = r(M) with one/both of the two facts: r( Im − ABA ) = m − r(ABA), r( Im − BAB ) = m − r(BAB). 2 h193i r(M ) = r(M) with one/both of the two facts: r( Im − ABA ) = m − Trace(ABA), r( Im − BAB ) = m − Trace(BAB). h194i r(M2) = r(M) with one/both of the two facts: BAB = A, ABA = B. 2 h195i r(M ) = r(M) with one/both of the two facts: A = PR(M)∩R(M∗) and B = PR(M)∩R(M∗). 208 Ë Yongge Tian

2 h196i r(M ) = r(M) and A + B = 2PR(M)∩R(M∗). 2 h197i r(M ) = r(M) and Im − A − B = −PR(M)∩R(M∗) + PR⊥(M)∩R⊥(M∗). 2 ∗ ⊥ ⊥ ∗ h198i r(M ) = r(M) and R( Im − A − B ) = (R(M) ∩ R(M )) ⊕ (R (M) ∩ R (M )). 2 h i ∗ ⊥ ⊥ ∗ 199 r(M ) = r(M) and PR( Im−A−B ) = PR(M)∩R(M ) + PR (M)∩R (M ). 2 m h200i r(M ) = r(M) and R(A − B) ⊕ R(Im − A − B) = C . 2 h201i r(M ) = r(M) with one/both of the two facts: AB = PR(M)∩R(M∗), BA = PR(M)∩R(M∗). 2 h202i r(M ) = r(M) and AB + BA = 2PR(M)∩R(M∗). h203i r(M2) = r(M) with one/both of the two facts: R(AB) = R(M) ∩ R(M∗), R(BA) = R(M) ∩ R(M∗). 2 h204i r(M ) = r(M) with one/both of the two facts: AB = PR(AB), BA = PR(BA). 2 h205i r(M ) = r(M) and AB + BA = PR(AB) + PR(BA). 2 ⊥ ⊥ ∗ h206i r(M ) = r(M) and R( Im − AB ) = R( Im − BA ) = R (M) ∩ R (M ). 2 h207i r(M ) = r(M) and Im − AB = PR⊥(M)∩R⊥(M∗). 2 h208i r(M ) = r(M) with one/both of the two facts: R(AB) ∩ R( Im − AB ) = {0}, R(BA) ∩ R( Im − BA ) = {0}. 2 m m h209i r(M ) = r(M) with one/both of the two facts: C = R(AB) ⊕ R( Im − AB ), C = R(BA) ⊕ R( Im − BA ). h i 2 210 r(M ) = r(M) with one/both of the two facts: Im = PR(AB) + PR( Im−AB ), Im = PR(BA) + PR( Im−BA ). 2 h211i r(M ) = r(M) with one/both of the two facts: ABA = PR(M)∩R(M∗), BAB = PR(M)∩R(M∗). 2 h212i r(M ) = r(M) and ABA + BAB = 2PR(M)∩R(M∗). h213i r(M2) = r(M) and R( AB + BA ) = R(M) ∩ R(M∗). h214i r(M2) = r(M) and R(AB) = R(BA). 2 m m h215i r(M ) = r(M) with one/both of the two facts: R(AB) ⊕ N (AB) = C , R(BA) ⊕ N (BA) = C . h216i r(M2) = r(M) with one/both of the two facts: R(AB) = R(A) ∩ R(B), R(BA) = R(A) ∩ R(B). h217i r(M2) = r(M) with one/both of the two facts: R(AeBe) = N (A) ∩ N (B), R(BeAe) = N (A) ∩ N (B). h218i r(M2) = r(M) and R(ABAe) = R(BABe).

Acknowledgements. The author would like to express his thanks to the referee for helpful comments and suggestions to an earlier version of this paper.

References

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