On Some Ordering Properties of the Generalized Inverses of Nonnegative Definite Matrices Chien-Fu Wu* Depa&ment of Statistics University of Wisconsin Madison, Wisconsin 53706 Submitted by Richard A. Bruakli ABSTRACT For any positive definite matrices A and B, it is known that A > B iff B - ’ >A - ‘. This paper investigates the extensions of the above result to any two real nonnegative definite matrices A and B. 1. INTRODUCTION For any real positive definite matrices A and B, it is known that A > B iff B - ’ >A - ‘. The extension of this result to real nonnegative definite (n.n.d.) matrices A and B is not trivial, since the “inverse” of A or B may not be uniquely defined. In this paper we consider the following question: for any n.n.d. matrices A and B with A > B and any given generalized inverse A - of A (or B - of B), does there exist a generalized inverse B - of I? (or A - of A) such that B - > A -? This question is answered for every possible combina- tion of the ranks of A,B,A-,B-, and A-,B- can be interpreted as any of the {I}-, {2}-, {I,2}-, or { 1,2,3,4}-inverses of A, B. All the matrices considered are real. Before stating the main results of the paper, we need to give some basic facts about generalized inverses. For any matrix A of real elements, a generalized inverse X is defined by one or several of the following conditions due to Penrose [5]: AxA=A, (1) xAx=x, (2) (AX)‘=AX, (3) (xA)‘=xA, (4) *Research supported by Research Committee, University of Wisconsin, Madison, and National Science Foundation Grant No. MCS-79-01846. LINEAR ALGEBRA AND ZTSAPPLZCATZONS32:49-60(1980) 49 0 Elsevier North Holland, Inc., 1980 @324-3795/80/040049+ 12$01.75 50 CHIEN-FU WU where A’ denotes the transpose of A. Following Ben-Israel and Greville [2], a matrix X is called a {i,i,..., I }-inverse of A if it satisfies the equations (i), ( i), . , (1) from among the equations (l), (2), (3), (4), and it is also denoted by A(‘*i*‘..,‘). For example, a { 1}-inverse is called a generalized inverse and a { 1,2}-inverse is called a reflexive generalized inverse in Rao and Mitra [7]; the { 1,2,3,4}-inverse (which is unique) is usually called the pseudo or Moore-Penrose inverse and also denoted by A+. For any {l}-, {2}-, or { 1,2}-inverse, it is well known that rank A(‘) > rank A, rank A@) < rank A, and rank A(‘,2!=rank A. For any n.n.d. matrix A, an A(‘) or Ac2) may not even be symmetric. From Theorem 1 of Sec. 2, any symmetric {2}-inuerse of an n.n.d. matrix A must be n.n.d. However, the corresponding result for { 1}-inverse does not hold. A characterization of the n.n.d. { 1}-inverses of any n.n.d. matrix A is given in Theorem 2. With this in mind, we will a.ssume that all the generalized inverses of an n.n.d. matrix considered in this paper are n.n.d. For any two n.n.d. matrices A and B, we define A > B iff A - B is n.n.d. Let the rank of A, B,A(‘), B(l) be denoted by r, s, i;,g. In order that A > B and B(l) >A(‘) hold, it is necessary that S>T>T>S. (5) In Theorem 5, we show that: (i) for any (?,g satisfying (5) and any A(‘) of rank i; there exists a B(l) of rank B such that B(l) >A(‘); (ii) if T = s, then for any (?: 8) satisfying (5) and any B (‘) of rank S, there exists an A(‘) of rank ? such that B(l) >A(‘). However, (ii) is not true for r >s. A class of counterex- amples to (ii) is given in Theorem 5A. Similarly, a necessary condition for A > B and Bc2) >AC2) is T>S>i>r, (6) where ?= rankAC2), S = rank B (2). In Theorem 6, we show that: (i) for any (T, q satisfying (6) and any B (2) of rank 8, there exists an A @) of rank ? such that Bc2) >AC2), (ii) if T= s, th en f or any (?,S) satisfying (6) and any AC2’ of rank 7, there exists a B c2) of rank S such that B c2)>Ac2). Again, (ii) is not true for T >s. For the special case A = B + xx’, a necessary and sufficient condition for (ii) to hold is x’AC2k < 1 (see Theorem 6A and Lemma 5). For a { 1,2}- or { 1,2,3,4} -inverse, a necessary condition on the ranks becomes ? = S = r = S. Results analogous to Theorems S(ii) and 6(ii) can be established. These are summarized in Theorems 7 and 8. In the course of proving these theorems, we have also obtained some other results of independent interest. In Theorem 3, we show that for any { 1}-inverse A (‘) of A and any u >T>D>T, r=rank A, ?=rank A(‘), there exist { 1}-inverses A:) and AC) of A with ranks u and u such that A$ > A(‘) ORDERING OF GENERALIZED INVERSES 51 >A:). A similar result for {2}-inverses is stated in Theorem 4. Lemmas 5 and 6 may also be useful in other contexts. They do not seem to exist in the literature. The question originally arose in the context of a statistical problem. For its applications in statistics, see Cheng and Wu [3] and Milliken and Akdeniz [41. 2. SOME RESULTS CONCERNING THE NONNEGATIVE DEFINITE GENERALIZED INVERSES OF A NONNEGATIVE DEFINITE MATRIX For any n X n matrix A, we denote the column space of A by %(A). A rank (or full rank) factorization of A with rank T is A = FG’, where F and G are both n X r matrices of rank T. The identity matrix of order r is denoted by ZI* LEMMA 1. Let A=QQ’ b e a rank factmizaticm of an nX n n.n.d. matrix A with rank r. Then G is a symmetric { 1,2}-inverse of A iff G cun be expressed as G = HH’, where H is an n X T matrixand H’ Q = Z,. In particu- lar, any symmetric {1,2}-inverse of an n.n.d. matrix is n.n.d. Proof. This can be proved simply by modifying the argument of Lemma 2.5.2 of Rao and Mitra [7, p. 281. It can also be found in Pringle and Rayner [6, p. 251. w THEOREM 1. Any symmetric {2}-inverse A(‘) of an n.n.d. matrix A is n.n.d. Proof. Since A = PP’ for some matrix P and A(‘)’ = AC2), AC2)= AC2)AAc2) = (AC2)P)(AC2)P)’ shows that A(‘) is n.n.d. H Since any n X n n.n.d. matrix A of rank r, r<n, can always be expressed as p’r op‘ [ 00’ 1 where P is an n x n nonsingular matrix, it follows from a result due to R. C. Bose [6, p. 81 that any symmetric { l}-inverse A(‘) of A can be expressed as (p’)-1 4 u p-l, [ U’ w 1 52 CHIEN-FU WU where Wand WarearbitraryrX(n-r)and(n-r)x(n-r)matrices,and W is symmetric. It is clear that the symmetric A(‘) is n.n.d. iff the middle matrix in (8) is n.n.d., which is not always true. Therefore, no result analogous to Theorem 1 holds for {l}- inverse. Later on, a different characterization of the n.n.d. { 1}-inverses will be needed. This is stated as THEOREM 2. For any n.n.d. matrix A of rank r, A(‘) is n.n.d. with rank s iff A(‘) is a symmetric {1,2}-inuerse of A + XX’, where X is of order n X (s - r) and rank(A :X) = s. Proof. The “if’ part follows from Theorem 2.7.1 of Rao and Mitra [7, p. 311. For the “only if’ part, we give an algebraic proof. From (8), any n.n.d. A(‘) of rank s can be expressed as py( ;, g)P-l, where is a rank factorization with F’F = I,., and F and G are s X r and s X (n - r) matrices with rank(F: G)=s. Let F, be an s X(s- r) matrix such that Fi F, = I,_ I and F’Fz = 0. Then, from (F: F,)‘( F : F2) = Z,, we have (F:F,)(F:F,)‘=z,=FF’+F,F;. (10) (i) First we want to show that there exists an n X (s- r) matrix X such that ( E~)(F:G)=(( i ~)+XX’)(LS2) and rank[ (i) i x]=s. (11) Since rank (F : G) = s, the equation F,=(F:G)X (12) is consistent. Any solution X can be expressed as X=(F:G)(‘)F,. (13) ORDERING OF GENERALIZED INVERSES 53 From (12), (13), and F’F, =O, it follows that x’ (14) and (Z,:F’G)X= F’(F:G)X=O. (15) From (lo), (12), (14), (15), it can be easily verified that the X defined in (13) satisfies (11). (ii) From Theorem 1.8 of Pringle and Rayner [6, p. 131, if P1 and Pz are nonsingular, then Pz-lA(‘%‘lwl is also a { 1,2}-inverse of P,AP,. Therefore, from (9) and (ll), the n.n.d.