RANK FACTORIZATION AND MOORE-PENROSE INVERSE Gradimir V. Milovanovic´ and Predrag S. Stanimirovic´ Abstract. In this paper we develop a few representations of the Moore- Penrose inverse, based on full-rank factorizations of matrices. These rep- resentations we divide into the two different classes: methods which arise from the known block decompositions and determinantal representation. In particular cases we obtain several known results. 1. Introduction Cm×n The set of m × n complex matrices of rank r is denoted by r = {X ∈ Cm×n |r : rank(X) = r}. With A and A|r we denote the first r columns of A and the first r rows of A, respectively. The identity matrix of the order k is denoted by Ik, and O denotes the zero block of an appropriate dimensions. We use the following useful expression for the Moore-Penrose generalized inverse A†, based on the full-rank factorization A = PQ of A [1-2]: A† = Q†P † = Q∗(QQ∗)−1(P ∗P )−1P ∗ = Q∗(P ∗AQ∗)−1P ∗. We restate main known block decompositions [7], [16-18]. For a given Cm×n matrix A ∈ r there exist regular matrices R, G, permutation matrices E, F and unitary matrices U, V , such that: 1991 Mathematics Subject Classification. 15A09, 15A24. Typeset by AMS-TEX 1 2 G. V. Milovanovi´cand P. S. Stanimirovi´c Ir O B O (T1) RAG = O O = N1, (T2) RAG = O O = N2, I K I O (T ) RAF = r = N , (T ) EAG = r = N , 3 O O 3 4 K O 4 Ir O Ir O (T5) UAG = O O = N1, (T6) RAV = O O = N1, B O B K (T7) UAV = O O = N2, (T8) UAF = O O = N5, B O (T ) EAV = = N , 9 K O 6 A11 A12 (T10) EAF = = N7, where rank(A11)=rank(A). A21 A22 (T11) Transformation of similarity for square matrices [11]: −1 ∗ −1 Ir K ∗ −1 T1 T2 RAR = RAF F R = O O F R = O O . The block form (T10) can be expressed in two different ways: A11 A11T (T10a) EAF = , where the multipliers S and T satisfy SA11 SA11T −1 −1 T = A11 A12, S = A21A11 (see [8]); A11 A12 A11 A12 (T10b) EAF = = −1 (see [9]). A21 A22 A21 A21A A12 11 Block representations of the Moore-Penrose inverse is investigated in [8], [11], [15–19]. In [16], [18] the results are obtained by solving the equations (1)–(4). In [15], [8] the corresponding representations are obtained using the block decompositions (T10a) and (T10b) and implied full-rank factorizations. Also, in [19] is introduced block representation of the Moore-Penrose in- verse, based on A† = A∗T A∗, where T ∈ A∗AA∗{1}. Block decomposition (T11) is investigated in [11], but only for square matrices and the group inverse. The notion determinantal representation of the Moore-Penrose inverse of A means representation of elements of A† in terms of minors of A. Deter- minantal representation of the Moore-Penrose inverse is examined in [1–2], [4–6], [12–14]. For the sake of completeness, we restate here several nota- tions and the main result. For an m × n matrix A let α = {α1,... ,αr} and β = {β1,... ,βr} be subsets of {1,... ,m} and {1,... ,n}, respectively. RANK FACTORIZATION AND MOORE-PENROSE INVERSE 3 α ... α A 1 r Aα A Then β1 ... βr = | β | denotes the minor of determined by the rows α indexed by α and the columns indexed by β, and Aβ represents the corre- α sponding submatrix. Also, the algebraic complement of Aβ is defined by ∂ α ... α i α ... α α1 ... αp−1 αp+1 ... αr | Aα |=A 1 p−1 p+1 r =(−1)p+qA . β ij β1 ... βq 1 j βq+1 ... βr β1 ... βq−1 βq+1 ... βr ∂aij − Adjoint matrix of a square matrix B is denoted by adj(B), and its deter- minant by |B|. Determinantal representation of full rank matrices is introuced in [1], and for full-rank matrices in [4–6]. In [12–14] is introduced an elegant derivation for determinantal representation of the Moore-Penrose inverse, using a full- rank factorization and known results for full-rank matrices. Main result of these papers is: Proposition 1.1. The (i, j)th element of the Moore-Penrose inverse G = Cm×n (gij ) of A ∈ r is given by α ∂ α Aβ | Aβ | α:j∈α ; β:i∈β ∂aji gij = P γ γ . | A δ | | Aδ | γ,δ P In this paper we investigate two different representations of the Moore- Penrose inverse. The first class of representations is a continuation of the papers [8] and [15]. In other words, from the presented block factorizations of matrices find corresponding full-rank decompositions A = PQ, and then ap- ply A† = Q∗(P ∗AQ∗)−1P ∗. In the second representation, A† is represented in terms of minors of the matrix A. In this paper we describe an elegant proof of the well-known determinantal representation of the Moore-Penrose inverse. Main advantages of described block representations are their simply derivation and computation and possibility of natural generalization. Deter- minantal representation of the Moore-Penrose inverse can be implemented only for small dimensions of matrices (n ≤ 10). 2. Block representation Cm×n Theorem 2.1. The Moore-Penrose inverse of a given matrix A ∈ r can be represented as follows, where block representations (Gi) correspond to the block decompositions (Ti), i ∈ {1,... , 9, 10a, 10b, 11}: ∗ −1 ∗ † −1 ∗ −1|r −1 ∗ −1|r (G1) A = G |r R A G |r R , 4 G. V. Milovanovi´cand P. S. Stanimirovi´c ∗ −1 ∗ † −1 ∗ −1|r −1 ∗ −1|r (G2) A = G |r R B A G |r R B , ∗ −1 ∗ I |r I |r (G ) A† = F r R−1 AF r R−1 , 3 K∗ K∗ −1 † −1 ∗ ∗ −1 ∗ ∗ (G4) A = G |r [ Ir, K ] EA G |r [ Ir, K ] E, −1 † −1 ∗ −1 ∗ (G5) A = G |r U|rA G |r U|r, ∗ −1 ∗ † |r −1|r |r −1|r (G6) A = V R AV R , † |r ∗ |r −1 ∗ (G7) A = V B U|rAV B U|r, −1 B∗ B∗ (G ) A† = F U AF U , 8 K∗ |r K∗ |r † |r ∗ ∗ |r −1 ∗ ∗ (G9) A = V [ B , K ] EAV [ B , K ] E, −1 Ir Ir (G ) A† = F A∗ [ I , S∗ ] EAF A∗ [ I , S∗ ] E 10a T ∗ 11 r T ∗ 11 r I = F r (I + T T ∗)−1A−1(I + S∗S)−1 [ I , S∗ ] E, T ∗ r 11 r r 1 A∗ A∗ − (G ) A† =F 11 (A∗ )−1[A∗ , A∗ ]EAF 11 (A∗ )−1[A∗ , A∗ ]E 10b A∗ 11 11 21 A∗ 11 11 21 12 12 A∗ =F 11 (A A∗ +A A∗ )−1A (A∗ A +A∗ A )−1[A∗ , A∗ ]E, A∗ 11 11 12 12 11 11 11 21 21 11 21 12 ∗ −1 ∗ † ∗ Ir −1|r ∗ Ir −1|r (G11) A =R −1 ∗ R T1 AR −1 ∗ R T1 . T1 T2 T1 T2 Proof. (G1) Starting from (T1), we obtain O −1 Ir −1 −1 Ir O −1 A = R O O G = R O [ Ir, ] G , which implies −1 Ir −1|r O −1 −1 P = R O = R , Q = [ Ir, ] G = G |r. Now, we get ∗ −1 ∗ † ∗ ∗ ∗ −1 ∗ −1 ∗ −1|r −1 ∗ −1|r A = Q (P AQ ) P = G |r R A G |r R . RANK FACTORIZATION AND MOORE-PENROSE INVERSE 5 The other block decompositions can be obtained in a similar way. (G5) Block decomposition (T5) implies O ∗ Ir −1 ∗ Ir O −1 A = U O O G = U O [ Ir, ] G , which means ∗ Ir ∗|r O −1 −1 P = U O = U , Q = [ Ir, ] G = G |r. (G7) It is easy to see that (T7) implies O ∗ B ∗ ∗ B O ∗ A = U O O V = U O [ Ir, ] V . Thus, ∗ B ∗|r O ∗ ∗ P = U O = U B, Q = [ Ir, ] V = V |r, which means ∗ ∗ ∗ |r P = B U|r, Q = V . (G10a) From (T10a) we obtain ∗ A11 A11T ∗ ∗ Ir ∗ A = E F = E A11 [ Ir, T ] F , SA11 SA11T S which implies, for example, the following full rank factorization of A: I P = E∗ r A , Q = [ I , T ] F ∗. S 11 r Now, 1 I , I , − A† = F r A∗ [ I , S∗ ] EAF r A∗ [ I , S∗ ] E. T ∗ 11 r T ∗ 11 r The proof can be completed using I EAF = r A [ I , T ] . S 11 r −1 Remarks 2.1. (i) A convenient method for finding the matrices S, T and A11 , required in (T10a) was introduced in [8], and it was based on the following extended Gauss-Jordan transformation: 6 G. V. Milovanovi´cand P. S. Stanimirovi´c A A I I T A−1 11 12 → 11 . A21 A22 O O O −S (ii) In [3] it was used the following full-rank factorization of A, derived A11 A12 from A = −1 : A21 A21A A12 11 A11 −1 P = , Q = [ Ir, A11 A12 ] . A21 3. Determinantal representation In the following definition are generalized concepts of determinant, alge- braic complement, adjoint matrix and determinantal representation of gen- eralized inverses. (see also [14].) Definition 3.1. Let A be m × n matrix of rank r. (i) The generalized determinant of A, denoted by Nr(A), is equal to α α Nr(A)= Aβ | Aβ | , α,β P (ii) Generalized algebraic complement of A corresponding to aij is † α ∂ α Aij = Aβ | Aβ | . α:j∈α;β:i∈β ∂aji P † (iii) Generalized adjoint matrix of A, denoted by adj (A) is the matrix † whose elements are Aij.