MOORE-PENROSE INVERSE IN AN INDEFINITE INNER PRODUCT SPACE

J. Appl. Math. & Computing Vol. 19(2005), No. 1-2, pp. 297-310 MOORE-PENROSE INVERSE IN AN INDEFINITE INNER PRODUCT SPACE K. KAMARAJ AND K. C. SIVAKUMAR Abstract. The concept of the Moore-Penrose inverse in an indefinite inner product space is introduced. Extensions of some of the formulae in the Euclidean space to an indefinite inner product space are studied. In particular range-hermitianness is completely characterized. Equivalence of a weighted generalized inverse and the Moore-Penrose inverse is proved. Finally, methods of computing the Moore-Penrose inverse are presented. AMS Mathematics Subject Classification : 15A09, 15A23. Key words and phrases : Moore-Penrose inverse, group inverse, range- Hermitian matrices, indefinite inner product. 1. Introduction An indefinite inner product in C n is a conjugate symmetric sesquilinear form [x, y] which satisfies the regularity condition: [x, y] =0, for each y C n holds only when x = 0. Any indefinite inner product is associated with a unique invertible hermitian matrix P n with complex entries such that [x, y] = x, P n y, where.,. denotes the Euclidean inner product on C n. The converse is also true. The study of linear transformations over indefinite inner product spaces has received a lot of attention over the past two decades. Several important structural characterizations have been established for many problems ([2], [9]) and the references cited therein. In this paper we study the notion of generalized inverses of matrices in an indefinite inner product space. We organize the paper as follows. In section 2, we give the known definitions for ready reference and some properties which are used throughout the paper. In section 3, we define the Moore-Penrose inverse in an indefinite inner product space and prove the existence and uniqueness. Here we consider special types of inverses like left (right) inverses and other subclasses of generalized inverses. In Section 4, we investigate some of the properties of the Received April 30, 2004. Revised August 6, 2004. Corresponding author. c 2005 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 297

298 K. Kamaraj and K. C. Sivakumar Moore-Penrose inverse in an IIPS. Here, we prove the equivalence of the Moore- Penrose inverse and the weighted generalized inverse. Finally, we consider the question of complementarity of the range space and the null space of the adjoint of a linear map. In section 5, we define range-hermitianness and give some equivalent conditions for matrices to be range-hermitian. In the final section we list some methods for computing the Moore-Penrose inverse in an indefinite inner product space. 2. Preliminary concepts We call a complex (real) square matrix A hermitian (symmetric) if A = A (A = A T ) and normal if AA = A A(AA T = A T A), where X denotes the complex conjugate transpose of the matrix X with respect to the Euclidean inner product.,. on C n. Let N be an invertible hermitian matrix of order n. An indefinite inner product in C n is defined by the equation [x, y] = x, Ny, where x, y C n. Such a matrix N is called a weight. A space with an indefinite inner product is called an indefinite inner product space (IIPS). We call u and v orthogonal if [u, v] = 0, where u, v C n. Let M and N be weights of order m and n, respectively. The MN adjoint of an m n matrix A denoted A [ ] is defined by A [ ] = N 1 A M. Sun and Wei [10] use the terminology weighted conjugate transpose for MN adjoint. We call a square matrix N hermitian (N normal) if A [ ] = A(AA [ ] = A [ ] A). In the Euclidean space setting, the Moore-Penrose inverse A of an m n complex matrix A is the unique solution to the equations AXA = A, XAX = X, (AX) = AX and (XA) = XA. It is well known that A exists for all matrices A [1]. The group inverse A # of square matrix A is the unique solution to the equations AXA = A, XAX = X and AX = XA. It is well known that A # exists if and only if rank(a) =rank(a 2 ). It can be shown that A is range hermitian, i.e., R(A) =R(A ) if and only if A = A # [1], where R(X) denotes the range space of X. A square matrix A is called a projection if A 2 = A and an N orthogonal projection if A is a projection and A [ ] = A. The next result gives the properties of adjoint that are easy to prove.

Moore-Penrose inverse in an indefinite inner product space 299 Proposition 1. Let A, B, and C be complex matrices of orders m p, p n and m p, respectively. Then (i) (A [ ] ) [ ] = A, (ii) (AB) [ ] = B [ ] A [ ], (iii) (A + C) [ ] = A [ ] + C [ ], (iv) If C n is an indefinite inner product space with weight N, then N [ ] = N. ( ) 1 0 Let N = G n =, where I 0 I n denotes the identity matrix of order n 1 n. Throughout the paper in all examples, N will denote G n of an appropriate size. 3. Existence and uniqueness of the Moore-Penrose inverse We start with a definition of Moore-Penrose inverse in an indefinite inner product space. Let A be an m n complex matrix. For an n m matrix X consider the following equations: AXA = A (1) XAX = X (2) (AX) [ ] = AX (3) (XA) [ ] = XA. (4) Note that I [ ] = I so that if A is invertible then by setting X = A 1, X clearly satisfies the above four equations. If X satisfies the four equations as above, then it can be shown to be unique and such an X will be denoted by A [ ]. Unlike the Euclidean ( case, ) a matrix need not have a Moore-Penrose inverse in an IIPS. 1 1 If A = with M = N = G 1 1 2, then it can be shown that there is no X satisfying equations (1)-(4). Let F be a field and F m n denote the set of all m n matrices over F. Let be any involution on F m n. Kalman [4] has shown that for A F m n A exists iff rank(a) =rank(a A)=rank(AA ). Our next result is just a particular case of his result in an IIPS. Theorem 1. Let A be an m n matrix. Then A [ ] exists iff rank(a) = rank(aa [ ] )=rank(a [ ] A). IfA [ ] exists, then it is unique. Proof. Uniqueness is as in the Euclidean case. ( ) 1 1 Remark 1. For the matrix A = mentioned above, we have AA 1 1 [ ] = 0 = A [ ] A. Thus rank(a) rank(aa [ ] ) and hence A [ ] does not exist.

300 K. Kamaraj and K. C. Sivakumar Proposition 2. Let x C n. Then x [ ] exists iff x [ ] x 0, in which case x [ ] = 1 x [ ] x x[ ]. Proof. Straightforward. More generally, we have the following: Theorem 2. Let A be an m n complex matrix of full-column rank. Then A ( [ ] 1A exists iff A [ ] A is invertible. In this case, A [ ] = A A) [ ] [ ]. Proof. Let A [ ] exist. Then rank(aa [ ] )=rank(a [ ] A)=rank(A), so that (A [ ] A) 1 exists. Conversely, if (A [ ] A) 1 exists, then X =(A [ ] A) 1 A [ ] can be verified to satisfy equations (1)-(4). Remark 2. If A is of full-column rank, A [ ] need not exist. Let A = 1 0 0 1. 1 0 ( ) 0 0 Then A [ ] A = and so rank(a) rank(a 0 1 [ ] A). Thus A [ ] does not exist. A consequence of Theorem 2 is as follows. Corollary 1. Let A be an m n complex matrix of full-row rank. Then A [ ] exists iff AA [ ] is invertible. In this case, A [ ] = A [ ] (AA [ ] ) 1. Proof. Replace A by A [ ] in Theorem 2. Remark 3. If A has a left ((right) inverse, A [ ] need not exist. Let A be as given 1 ) 1 in Remark 2, then X = 2 0 2 is a left inverse of A but A 0 1 0 [ ] does not exist. We conclude this section with results in an IIPS that are analagous to the subclasses A{1, 2, 3} and A{1, 2, 4}, see [11]. Theorem 3. Let A be an m n complex matrix. If rank(a [ ] A)=rank(A), then (A [ ] A) (1) A [ ] A{1, 2, 3} and if rank(aa [ ] )=rank(a), then A [ ] (AA [ ] ) (1) A{1, 2, 4}.

Moore-Penrose inverse in an indefinite inner product space 301 Proof. Since rank(a [ ] A)=rank(A), we have R(A [ ] A)=R(A [ ] ). Thus A [ ] = A [ ] AU for some U. Thus A = U [ ] A [ ] A. Set Y =(A [ ] A) (1) A [ ]. Then ( (1)A ( (1)A AY A = A A A) [ ] [ ] A = U [ ] A [ ] A A A) [ ] [ ] A = A. Thus rank(a) rank(y ), but rank(y ) rank(a [ ] )=rank(a). Therefore, rank(a)= rank(y ). Then Y A{1, 2}. Also ( ) ( (1)A AY = U [ ] A [ ] A (A [ ] A) (1) A [ ] = U [ ] A [ ] A A A) [ ] [ ] AU = U [ ] A [ ] AU, which is clearly M hermitian. Thus (A [ ] A) (1) A [ ] A{1, 2, 3}. The second part follows similarly. Theorem 4. Let A be an m n complex matrix such that A [ ] exists. Then A [ ] = A (1,4) AA (1,3), where A (1,3) and A (1,4) are any {1, 3} and {1, 4} inverses of A, respectively. Proof. Follows by a direct computation, using Theorem 3. The following theorem is a reformulation of a method that is given in [5]. We present it here with a simpler proof. Theorem 5. Let A be m n complex matrix such that A [ ] exists. Then A [ ] = A [ ] (AA [ ] ) (1) A(A [ ] A) (1) A [ ], where (AA [ ] ) (1) AA [ ] {1} and (A [ ] A) (1) A [ ] A{1}. Proof. By Theorem 3, A [ ] (AA [ ] ) (1) A{1, 4} and (A [ ] A) (1) A [ ] A{1, 3}. Then by using Theorem 4, we get the desired result. 4. Properties of A [ ] In this section, we first prove the equivalence of a weighted generalized inverse and the Moore-Penrose inverse in an IIPS. Definition 1. Let A be a matrix of order m n and let M and N be invertible hermitian matrices of orders m and n, respectively. An n m matrix X is said to be a weighted generalized inverse of A (with respect to M,N) if the following equations hold: AXA = A XAX = X

302 K. Kamaraj and K. C. Sivakumar (MAX) = MAX (NXA) = NXA. Remark 4. It is well known that, a weighted generalized inverse A MN of A exists iff rank(a) =rank(an 1 A )=rank(a MA) ([7], [8], [12]). This is equivalent to rank(a) =rank(aa [ ] )=rank(a [ ] A). Thus A [ ] exists iff A MN exists. Interestingly, in this case these two coincide. We prove this next. Theorem 6. Let A be an m n complex matrix. If M and N are hermitian invertible matrices, then A [ ] = A MN, if it exists. Also, A MN is unique. Proof. We have (AX) [ ] = M 1 (AX) M = M 1 (MAX). Thus (AX) [ ] = AX M 1 (MAX) = AX (MAX) = MAX. Similarly, (XA) [ ] = XA (NXA) = NXA. Hence A [ ] = A MN. Uniqueness of A MN follows from the uniqueness of A[ ]. Remark 5. It must be emphasized that the concept of weighted Moore-Penrose inverse was studied in [7] especially to extend the results in the Euclidean spaces to general inner product spaces where the inner product is not necessarily positive definite. In other words, over indefinite inner product spaces. In this sense, our result here for weighted Moore-Penrose inverses is not new. However, we present this for the sake of completeness. Also, our proof appears to be simpler. Remark 6. It is well known that if M and N are positive definite, then A MN ( ) exists for all A. In this case, A MN = N 1 2 M 1 2 AN 1 1 2 M 2 [10]. Thus A [ ] exists in all such cases, by the above result. In the Euclidean space setting, we know that (A ) =(A ). A similar result holds in an IIPS, as we prove next. Proposition 3. If A [ ] exists, then (A [ ] ) [ ] = ( A [ ] ) [ ]. Proof. Setting X =(A [ ] ) [ ], we have A [ ] XA [ ] = (AA [ ] A) [ ] = A [ ], XA [ ] X = (A [ ] AA [ ] ) [ ] =(A [ ] ) [ ], A [ ] X = A [ ] A, and XA [ ] = AA [ ].

Moore-Penrose inverse in an indefinite inner product space 303 Thus A [ ] X and XA [ ] are N hermitian and M hermitian, respectively. Thus X =(A [ ] ) [ ]. Theorem 7. Let A be an m n matrix. (A [ ] A) [ ] exist. In this case, (AA [ ] ) [ ] = ( A [ ] ) [ ]A [ ] and If A [ ] exists, then (AA [ ] ) [ ] and ( [ ] ( ) [ ]. A A) [ ] =A [ ] A [ ] Proof. Since A [ ] exists, rank(aa [ ] )=rank (AA [ ] AA [ ]) = rank ((AA [ ] ) 2 (A [ ] ) [ ] A [ ]) ( rank (AA [ ] ) 2). ) Thus rank(aa [ ] )=rank((aa [ ] ) 2 ). Similarly, rank(a [ ] A)=rank ((A [ ] A) 2. By Theorem 1, (AA [ ] ) [ ] and (A [ ] A) [ ] exist. By setting X =(A [ ] ) [ ] A [ ],we get AA [ ] XAA [ ] = AA [ ] (A [ ] ) [ ] A [ ] AA [ ] = AA [ ]. Similarly we can easily verify the other three equations for the Moore-Penrose inverse in an IIPS. Using Theorem 7 and Proposition 3, the following can be established. Corollary 2. Let A be an m n matrix such that A [ ] exists. Then A [ ] = A [ ]( AA [ ]) [ ] =(A [ ] A) [ ] A [ ]. Proof. By Theorem 7, (AA [ ] ) [ ] and (A [ ] A) [ ] exist. Setting X = A [ ] (AA [ ] ) [ ], by Theorem 3, X A{1, 2, 4}. Thus it is enough to show that X A{3}. Since AX = AA [ ] (AA [ ] ) [ ] = AA [ ] (A [ ] ) [ ] A [ ] = AA [ ] is M hermitian. Thus X A{3}. The second equation follows similarly. Remark 7. We note that the converse of Theorem 2 is not true in general. Let A be as given in Remark 1. Then AA [ ] = 0, so that (AA [ ] ) [ ] = 0 but A [ ] does not exist. Note that A [ ] (AA [ ] ) [ ] =(A [ ] A) [ ] A [ ] = 0. Let K and H be subspaces of an indefinite inner product space V such that K H = V. The projection of V on K along H is denoted by P K, H. The

304 K. Kamaraj and K. C. Sivakumar following example shows that R(A) and N (A [ ] ) are not complementary, in general. Example 1. Let A be as given in Remark 1. Then R(A) =N (A [ ] ), so that R(A) and N (A [ ] ) are not complementary. The following theorem gives a sufficient condition under which R(A) and N (A [ ] ) are orthogonal complementary subspaces. Theorem 8. Let A be an m n matrix such that A [ ] exists. Then R(A) and N (A [ ] ) are orthogonal complementary subspaces of C n. Proof. Let x R(A) and y N(A [ ] ). Then for some z, [x, y] =[Az, y] =[z,a [ ] y]=0. Thus R(A) and N (A [ ] ) are mutually orthogonal. Let x R(A) N(A [ ] ), then for some y C n, x = Ay = AA [ ] Ay = AA [ ] x =(A [ ] ) [ ] A [ ] x = 0. Thus R(A) N(A [ ] )={0}. Since rank(a) =rank(a [ ] ), the dimensions of N (A [ ] ) and N (A) are equal. Then by the rank nullity dimension theorem, R(A) N(A [ ] )=C n. This completes the proof. Theorem 9. If A [ ] exists, then R(A [ ] )=R(A [ ] ) and N (A [ ] )=N(A [ ] ). Also AA [ ] = P R(A),N (A [ ] ) and A [ ] A = P N (A [ ] ), R(A). Proof. Since A [ ] is a {1, 2}- inverse of A, rank(a) =rank(a [ ] )=rank(a [ ] ). Thus the dimensions of R(A [ ] ) and R(A [ ] ) are equal and so the dimensions of N (A [ ] ) and N (A [ ] ) are equal. Clearly, R(A [ ] )=R(A [ ] AA [ ] ) R(A [ ] ). Thus R(A [ ] )=R(A [ ] ). Similarly, N (A [ ] ) N(A [ ] (A [ ] ) [ ] A [ ] )=N(A [ ] ). Thus N (A [ ] )=N(A [ ] ). Clearly, AA [ ] is idempotent and M hermitian. Also R(AA [ ] )=R(A) and N (AA [ ] )=N(A [ ] )=N(A [ ] ). Thus AA [ ] is the M orthogonal projection on R(A) along N (A [ ] ). The second part follows similarly.

Moore-Penrose inverse in an indefinite inner product space 305 5. Range-Hermitian matrices Range-hermitian matrices are a special classes of square matrices for which the Moore-Penrose inverse and the group inverse coincide. In this section we study extensions of results for range-hermitian matrices to matrices over IIPS. Definition 2. Let A be an n n complex matrix. Then A is called N range hermitian if R(A) =R(A [ ] ). Theorem 10. If A be a complex square matrix. Then the following are equivalent: (i) A is N range hermitian. (ii) AN 1 is range hermitian. (iii) A = A [ ] U, for some nonsingular matrix U. (iv) A = QA [ ], for some nonsingular matrix Q. (v) N (A) =N (A [ ] ). (vi) NA is range hermitian. Proof. (i) (ii): It comes from R(A) =R(A [ ] ) R(AN 1 )=R((AN 1 ) ). (ii) (iii): Since AN 1 is range hermitian, there exists an invertible matrix K such that (AN 1 ) = KAN 1 [1]. This in turn is equivalent to A = A [ ] U, where U = K 1. (iii) (iv) and (iv) (v) Follow easily. (v) (vi): Proof is similar to (i) (ii). (vi) (i): R(NA)=R((NA) ) NA =(NA) K for some nonsingular matrix K A = A [ ] K. Thus R(A) =R(A [ ] ). This completes the proof. It is known in the Euclidean case that A A = AA iff R(A) =R(A ) [1]. In such a case A # = A. The next result gives an analogous result in an IIPS under the condition that A [ ] exists. Theorem 11. Let A be a complex square matrix such that A [ ] exists. Then R(A) =R(A [ ] ) iff AA [ ] = A [ ] A. Proof. Since R(A) =R(A [ ] ), by Theorem 9 AA [ ] = P R(A),N (A [ ] ) = P R(A [ ] ),N (A) = A [ ] A. Conversely, R(A) =R(AA [ ] )=R(A [ ] A)=R(A [ ] )=R(A [ ] ).

306 K. Kamaraj and K. C. Sivakumar Corollary 3. Let A be a complex square matrix of order n such that A [ ] exists. Then there exists a polynomial p such that A [ ] = p(a) iff R(A) =R(A [ ] ). Proof. Suppose that R(A) =R(A [ ] ). Since A [ ] exists, by Theorem 11 A [ ] A = AA [ ]. Since A [ ] A{1, 2}, we have A [ ] = A #. The conclusion follows immediately [7, p. 97]. Conversely, if there exists a polynomial p such that A [ ] = p(a), then it is clear that AA [ ] = A [ ] A. Thus by Theorem 11, R(A) =R(A [ ] ). Penrose [6] shows that, if A is a normal matrix then for all positive integers n, (A n ) =(A ) n. This has been generalized to range hermitian matrices [1]. This generalization holds in an IIPS, as we prove next. Theorem 12. Let A be a complex square matrix such that A [ ] exists. If R(A) = R(A [ ] ), then for every positive integer n, (A n ) [ ] exists and (A n ) [ ] =(A [ ] ) n. Proof. First we prove the existence of (A n ) [ ]. Since A [ ] exists and R(A) = R(A [ ] ), we have A [ ] = A # so that rank(a n )=rank(a n+1 ) for all integers n 1. Since A =(A [ ] ) n 1 A n, rank(a n ) = rank(a) =rank(aa [ ] ) ( ( = rank A [ ]) [ n 1 ( A n (A n ) [ ] A [ ]) ] ) n 1 [ ] ( rank A n (A n ) [ ]). Thus rank(a n ) = rank ( A n (A n ) [ ]). Similarly we prove rank(a n ) = rank ( (A n ) [ ] A n). Thus (A n ) [ ] exists. Since R(A) = R(A n ), we have R(A n ) R ( (A n ) [ ]). Thus R (A n ) = R((A n ) [ ] ). Then by Theorem 11, (A [ ] ) n = (A # ) n =(A n ) # =(A n ) [ ]. Remark 8. The assumption that A is N range hermitian in the above theorem is indispensable. Let A = 2 0 0 1 1 1. Then A [ ] exists and 1 1 1 (A 2 ) [ ] A 2 = 0. Sorank(A 2 ) rank ( (A 2 ) [ ] A 2). Thus (A 2 ) [ ] does not exist. Remark 9. It is well known that every normal matrix is range-hermitian in Euclidean case [1]. But an analogous statement does not hold in an IIPS. Let A, M and N be as given in Remark 1. Then A is N normal but not N range hermitian. However, if A [ ] is assumed to exist then we have an analogous result, as we show next.

Moore-Penrose inverse in an indefinite inner product space 307 Theorem 13. Let A be a N normal matrix. If A [ ] exists, then A is N range hermitian. Proof. Since A [ ] exists, R(A) =R(AA [ ] )=R(A [ ] A)=R(A [ ] ). ( ) 1 1 Remark 10. Let A =. Then A is N normal and N range 1 1 hermitian. But A [ ] does not exist, as can be verified. This example shows that the converse of Theorem 13 does not hold. 6. Methods for computing A [ ] Only very few algebraic methods are available in the literature for computing weighted generalized inverses, to the best of our knowledge ([5], [7] and [10]). Since A MN = A[ ] we suggest in this section, alternate methods. It will be clear that these methods work better. The following theorem is due to Decell [3]. This gives a method to calculate the Moore-Penrose inverse of a matrix in the setting of finite dimensional Euclidean spaces and uses the Cayley-Hamilton theorem. Theorem 14. Let A be any m n complex matrix and let f(x) =( 1) [x n n + ] γ 1 x n 1 + + γ n 1 x + γ n be the characteristic polynomial of AA. If k 0 is the largest integer such that γ k 0, then the Moore-Penrose inverse of A is given by ] A = (γ k ) 1 A [(AA ) k 1 + γ 1 (AA ) k 2 +... + γ k 2 AA + γ k 1 I. If γ k =0for all k, then A = 0. The next result is analogous to Decell s result for the Moore-Penrose inverse in an IIPS. Theorem 15. [ Let A be any m n complex matrix such that ] A [ ] exists and let f(x) =( 1) n x n + µ 1 x n 1 + µ 2 x n 2 +... + µ n 1 x + µ n be the characteristic polynomial of AA [ ]. If k 0is the largest integer such that µ k 0, then the Moore-Penrose inverse of A is given by [ ( A [ ] = (µ k ) 1 A [ ] AA [ ]) k 1 + µ1 (AA [ ]) ] k 2 +... + µk 2 AA + µ k 1 I. If µ k =0for all k, then A [ ] = 0.

308 K. Kamaraj and K. C. Sivakumar Proof. Since A [ ] exists, by Theorems 7 and 13, (AA [ ] ) [ ] exists and AA [ ] is M range hermitian. By Theorems 11 and 12, we have [ (AA [ ] ) [ ] AA [ ] = AA [ ] (AA [ ] ) [ ], and (AA [ ] ) n+1] [ ] (AA [ ] ) n =(AA [ ] ) [ ] for all positive integers n. By Cayely-Hamilton Theorem, (AA [ ] ) n + µ 1 (AA [ ] ) n 1 +... + µ n 1 AA [ ] + µ n I = 0. If k 0 is the largest [ integer such that µ k 0, then ] (AA [ ] ) n k (AA [ ] ) k + µ 1 (AA [ ] ) k 1 +... + µ k 1 AA [ ] + µ k I = 0. Pre-multiplying by [ (AA [ ] ) n k+1] [ ] we get, [ ] (AA [ ] ) [ ] (AA [ ] ) k + µ 1 (AA [ ] ) k 1 +... + µ k 1 AA [ ] + µ k I = 0, (AA [ ] ) k 1 + µ 1 (AA [ ] ) k 2 +... + µ k1 (AA [ ] ) [ ] AA [ ] + µ k (AA [ ] ) [ ] = 0. Pre-multiplying by A [ ] [ and using Theorem 2 we get ] A [ ] (AA [ ] ) k 1 + µ 1 (AA [ ] ) k 2 +... + µ k 1 I + µ k A [ ] = 0. This completes the proof. Example 2. Let A = 1 1 1 1 1 1. Then the characteristic polynomial of 1 1 1 1 1 1 AA [ ] is x 3 x 2. Thus k = 1 and µ k = 1. Thus A [ ] = 1 1 1. 1 1 1 Penrose [6] shows that the systems AA X = A and A AY = A are consistent. He gives a method to calculate the Moore-Penrose inverse by using the solutions of the above systems. We next explore this idea in an IIPS. Theorem 16. Let A be an m n matrix and X, Y be complex n m matrices. Then the systems AA [ ] X [ ] = A and A [ ] AY = A [ ] are consistent iff rank(a) = rank(aa [ ] )=rank(a [ ] A)(i.e., iff A [ ] exists). In this case A [ ] = XAY, where X and Y are solutions of AA [ ] X [ ] = A and A [ ] AY = A [ ], respectively. Proof. If the systems AA [ ] X [ ] = A and A [ ] AY = A [ ] are consistent, then rank(a) =rank(aa [ ] X [ ] ) rank(aa [ ] ) and rank(a) =rank(a [ ] )=rank(a [ ] AY ) rank(a [ ] A).

Moore-Penrose inverse in an indefinite inner product space 309 Thus rank(a) =rank(aa [ ] )=rank(a [ ] A). On the other hand, if A [ ] exists, then by Theorem 7 (AA [ ] ) [ ] exists and by Theorem 3, A [ ] (AA [ ] ) [ ] A{1, 2, 4}. Thus AA [ ] (AA [ ] ) (1) A = A. Then the system AA [ ] X [ ] = A is consistent. Similarly we can prove that the equation A [ ] AY = A [ ] is consistent. Let X and Y be defined as above. Pre multiplying by A [ ] and (A [ ] ) [ ], the systems AA [ ] X [ ] = A and A [ ] AY = A [ ], respectively we get A [ ] X [ ] = A [ ] A and AY = AA [ ]. Then A [ ] = A [ ] (AA [ ] )=(A [ ] A)Y = A [ ] X [ ] Y =(XA) [ ] Y = XAY. Example 3. Let A be as given in Example 2. Then X = 1 0 0 1 0 0 and 1 0 0 Y = 1 1 1 0 0 0 can be verified to be solutions of the systems AA [ ] X [ ] 0 0 0 1 1 1 = A and A [ ] AY = A [ ], respectively. Thus A [ ] = XAY = 1 1 1. 1 1 1 Corollary 4. Let A be an N hermitian matrix. Then the system A 2 X [ ] = A is consistent iff A [ ] exists. In this case A [ ] = XAX [ ], where X is a solution of A 2 X [ ] = A. Proof. Since A = A [ ], by Theorem 16, A [ ] exists iff A 2 X [ ] = A is consistent. Also, if X is a solution of A 2 X [ ] = A then A [ ] = XAX [ ]. Corollary 5. Let A be an m n matrix such that A [ ] exists. Then A [ ] = A [ ] XAA [ ] X [ ], where X is a solution of (AA [ ] ) 2 X [ ] = AA [ ]. Proof. Clearly, AA [ ] is M hermitian and (AA [ ] ) [ ] exists. Then by Corollary 4, (AA [ ] ) 2 X [ ] = AA [ ] is consistent and (AA [ ] ) [ ] = XAA [ ] X [ ]. Then using Theorem 2 we get A [ ] = A [ ] (AA [ ] ) [ ] = A [ ] XAA [ ] X [ ]. 7. Acknowledgements The authors thank the referee for the comments and suggestions which helped improve the presentation of the paper.

310 K. Kamaraj and K. C. Sivakumar References 1. A. Ben-Israel and T. N. E. Greville, Generalized Inverse: Theory and Applications, Wiley, New York, 1974. 2. J. Bognar, Indefinite Inner Product Spaces, Springer Verlag, 1974. 3. H.P. Decell, An Application of the Cayley - Hamilton Theorem to Generalized Matrix Inversion, SIAM. Rev. 7 (1965), 526-528. 4. R. E. Kalman, Alegebraic Aspects of the Generalized Inverse of a Rectangular Matrix, Generalized Inverses and Applications, (Ed. M.Z. Nashed), 111-123, Academic Press, 1976. 5. K. Manjunatha Prasad and R. B. Bapat, The Generalized Moore-Penrose inverse, Lin. Alge. Appl. 165 (1992), 59-69. 6. R. Penrose, A Generalized Inverse for Matrices, Proc. Camb. Philos. Soc. 51 (1955), 406-413. 7. C. R. Rao, Linear Statistical Inference and its Applications, Wiley, New York, 1973. 8. C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and its Appliations, Wiley, New York, 1971. 9. L. Rodman, Finite dimensional spaces with indefinite scalar products: Research problems, Obzornik za Matematiko in Fiziko 46 (1999), 57-60. 10. W. Sun and Y. Wei, Inverse Order Rule for Weighted Generalized Inverse, SIAM. J. Matrix Anal. Appl. 19 (1996), 772-775. 11. N. S. Urquhart, Computation of Generalized Inverse Matrices Which Satisfy Specified Conditions, SIAM Rev. 10 (1968), 216-218. 12. Xuchu He and Wenyu Sen, Introduction to Generalized Iverses of Matrices, Jiangsu Sci. & Tech. Publishing House, Nanjing, 1991. K. Kamaraj received his M. Sc. degree in Science from the College of Engineering, Anna University in 2001. Currently, he is a research student in Anna University. His research interests include matrix theory in general and in particular matrix generalized inverses and theoretical aspects of indefinite inner product space and their applications to quantum logics. Department of Mathematics, College of Engineering, Guindy, Anna University, Chennai 600 025, India e-mail: krajkj@yahoo.com K. C. Sivakumar received his Master of Science and Ph. D from Indian Institute of Technology (Madras). From 1995 to 2003 he was a lecturer in Anna university. Since May 2003 he is with Indian Institute of Technology (Madras). His research interests include operator theory and mathematical programing. Department of Mathematics, Indian Institute of Technology (Madras), Chennai 600 036, India e-mail: kcskumar@iitm.ac.in