Product of Range Symmetric Block Matrices in Minkowski Space

Transcription

1 BULLETIN of the Malaysian Mathematical Sciences Society Bull. Malays. Math. Sci. Soc. (2) 29(1) (2006), Product of Range Symmetric Block Matrices in Minkowski Space 1 AR. Meenakshi and 2 D. Krishnaswamy 1 Faculty of Engineering and Technology, Annamalai University, Annamalai Nagar , Tamil Nadu, India 2 Directorate of Distance Education, Annamalai University, Annamalai Nagar , Tamil Nadu, India 1 arm meenakshi@yahoo.co.in, 2 krishna swamy2004@yahoo.co.in Abstract. Necessary and sufficient conditions for the product of range symmetric matrices of rank r to be range symmetric in Minkowski space M is derived. Also equivalent conditions for the product of two range symmetric block matrices to be range symmetric are established. As an application we have shown that a block matrix in Minkowski space can be expressed as a product of range symmetric matrices in M Mathematics Subject Classification: Primary 15A57, Secondary 15A09 Key words and phrases: Minkowski space, range symmetric matrix 1. Introduction Throughout we shall deal with C n n, the space of n n complex matrices. Let C n be the space of complex n-tuples, we shall index the components of a complex vector in C n from 0 to n 1, that is u (u 0, u 1, u 2,... u n 1 ). Let be the Minkowski metric tensor defined by u (u 0, u 1, u 2,..., u n 1 ). Clearly the Minkowski metric matrix 1 0 (1.1), and 2 I 0 I n. n 1 In 8, Minkowski inner product on C n is defined by (u, v) u, v, where.,. denotes the conventional Hilbert(unitary) space inner product. A space with Minkowski inner product is called a Minkowski space and denoted as M. For A C n n, x, y C n, by using (1.1), (1.2) (Ax, y) Ax, y x, A y x, (A )y x, A y (x, A y) where A A. The matrix A is called the Minkowski adjoint of A in M (A is usual hermitian adjoint of A ). Naturally, we call a matrix A C n n Received: November 18, 2004; Revised: March 9, 2005.

2 60 AR. Meenakshi and D. Krishnaswamy M-symmetric in M if A A. From the definition A A we have the following equivalence: A is M-symmetric A is hermitian A is hermitian. For A C n n, rk(a), N(A), and R(A) are respectively the rank of A, null space of A and range space of A. By a generalized inverse of A we mean a solution of the equation A A and is denoted as A (1). A{1} is the set of all generalized inverses of A. Throughout I refers to identity matrix of appropriate order unless otherwise specified. Definition , Definition 1, p.7 For A C m n, A + is the Moore-Penrose inverse of A if AA + A A, A + AA + A +, AA + and A + A are Hermitian. The Minkowski inverse of A, analogous to Moore-Penrose inverse of A is introduced and its existence is discussed in 5. Definition , Definition 4, p.2 For A C m n, A m is the Minkowski inverse of A if AA m A A, A m AA m A, AA m and A m A are M-symmetric. Theorem , Theorem 1, p.4 For A C m n, A m exists in M rk(a) rk(aa ) rk(a A). Theorem , Lemma 3.3, p.143 Let A and B be matrices in M. N(A ) N(B ) N(A ) N(B ). Then Theorem , Lemma 1, p.193 For A, B, C C m n, the following are equivalent: (1) CA (1) B is invariant for every A (1) C nxm. (2) N(A) N(C) and N(A ) N(B ) 3. C CA (1) A and B AA (1) B for every A (1) A{1}. Theorem , Lemma 2.3, p.139 For A 1, A 2 C n n (A 1 A 2 ) A 2 A 1 (A 1 ) A 1. and A matrix A C n n is said to be range symmetric in unitary space (or) equivalently A is said to be EP if N(A) N(A ) (or AA + A + A) 2, p.163. For further properties of EP matrices one may refer 1,2, 4 and 9. In 6 the concept of range symmetric matrix in M is introduced and developed analogous to that of EP matrices. A matrix A C n n said to be range symmetric in M N(A) N(A ). In the sequel we shall make use of the following results. Theorem , Theorem 2.2, p.47 For A C n n, the following are equivalent: (i) A is range symmetric in M (ii) A is EP (iii) A is EP (iv) N(A ) N(A) (v) R(A) R(A ) (vi) A HA AK for some non-singular matrices H and K. (vii) R(A ) R(A)

3 Product of Range Symmetric Block Matrices in Minkowski Space Product of range symmetric matrices in M In this section we have obtained necessary and sufficient conditions for the product of two range symmetric matrices of rank r to be range symmetric in M. Later we have extended the result to block matrices in M. Theorem 2.1. Let A and B be range symmetric matrices of rank r in M and AB be of rank r. Then AB is range symmetric in M if and only if R(A) R(B). Proof. Let A and B be range symmetric matrices of rank r in M. Let AB be of rank r and R(A) R(B). We prove that AB is range symmetric in M. Clearly R(AB) R(A). Since rk(ab) rk(a) r, it follows that R(AB) R(A). Also R(AB) R(B ) R(B) and rk(ab) rk(ab) rk(a) rk(b) r. This implies R(AB) R(B). Since R(A) R(B), if follows that R(AB) R(AB). Hence AB is range symmetric in M. Conversely, AB is range symmetric in M implies R(AB) R(AB). R(AB) R(A) and rk(ab) rk(a) r implies R(AB) R(A). R(AB) R(B ) R(B). Thus R(A) R(B). Hence forth we are concerned with n n matrices M partitioned in the form A B (2.1) M with rk(m) rk(a) r C D It is well known that in 3 M of the form (2.1) satisfies N(A) N(C), N(A ) N(B ) and D CA + B. Definition , Lemma 1.2, p.193 Let A B M C D be an n n matrix. The Schur complement of A in M, denoted by M/A is defined as D CA (1) B, where A (1) is a generalized inverse of A. Theorem 2.2. Let M be of the form (2.1), then M is range symmetric in M A is range symmetric in M and CA + 1 (A + B), where 1 is the Minkowski metric tensor of order as that of A. Proof. Since A is range symmetric in M and CA + 1 (A + B), by Theorem 1.5 (ii), 1 A is EP and ( 1 A) + A + 1. Hence 1 A( 1 A) + ( 1 A) + 1 A. Since + 1 1, 1 AA + 1 A A. By (1.1) for 1, we have (2.2) 1 AA + 1 A + A. Since rk(m) rk(a) r, we have N(A) N(C), N(A ) N(B ) and Schur complement of A in M is zero. By Theorem 1.3, we have C CA + A, B AA + B and D CA + B. Let us consider the matrices P I 0 CA + I, Q I A + B 0 I and L A 0,

4 62 AR. Meenakshi and D. Krishnaswamy where P, Q are non-singular. Now I 0 A 0 P LQ CA + I A AA + B CA + A CA + B A B C D M I A + B 0 I and by using CA + 1 (A + B), we have Q Q 1 0 I 0 0 I (A + B) I I 0 (A + B) 1 I I 0 1 (A + B) I I 0 CA + I P I Since A is range symmetric in M, L is range symmetric in M. We claim M is range symmetric in M, that is N(M) N(M ). Since M P LP, x N(M) Mx 0 P LP x 0 Ly 0 where y P x L y 0 by N(L) N(L ) P L P x 0 M x 0 x N(M ). Thus N(M) N(M ) and hence M is range symmetric in M. Conversely, let us assume that M is range symmetric in M. Since M P LQ, one choice of M (1) is M (1) Q 1 A + 0 P 1. Since M is range symmetric in M, we have N(M) N(M ). By using Theorem 1.2, and Theorem 1.3 we get M M M (1) M. By using (1.2), we have M

5 Product of Range Symmetric Block Matrices in Minkowski Space 63 M M (1) M and by using (1.1), M M M (1) M M Q 1 A + 0 P 1 A 0 P Q M Q 1 A + A 0 Q M I (A + B) A + A 0 0 I M A + A A + B A C B D A C 1 0 A + A A + B B D 0 I A 1 A + A 1 A 1 A + B B 1 A + A 1 B 1 A + B Equating the corresponding blocks, we get I (A + B) 0 I I A A 1 A + A 1 1 A 1 1 A 1 A + A A A A + A. By Theorem 1.3, it follows that N(A) N(A ) and rk(a) rk(a ). N(A) N(A ) and therefore A is range symmetric in M. Also C A 1 A + B 1 C 1 1 A 1 A + B 1 C A A + B 1. Hence Taking Minkowski adjoint, and by using 1 1 and Theorem 1.4, we get C 1 (A + B) A. Now CA + 1 (A + B) AA + 1 (A + B) 1 A + A 1 1 (A + B) 1 (A + A) 1 By using (2.2) 1 (A + B) (A + A) 1 (A + AA + B) By using Theorem (A + B). This completes the proof. Lemma 2.1. Let M be of the form (2.1) be range symmetric in M, then A is range symmetric in M and there exists an r (n r) matrix X such that A (2.3) M 1 X A 1 X where 1 is the Minkowski metric tensor of order as that of A. Proof. Since M is of the form (2.1) by using Theorem 1.2 and Theorem 1.3, M satisfy N(A) N(C), N(A ) N(B ) and D CA + B. Hence there exist (n r) r matrix Y and n (n r) matrix X such that C Y A and B.

6 64 AR. Meenakshi and D. Krishnaswamy Since A is range symmetric in M by using Theorem 1.5(ii). 1 A is EPr, again by using (2.2) we have AA + 1 A + A 1, CA + 1 (A + B), and Y AA + 1 (A + ) 1 X (A + A) By using Theorem X 1 (A + A) 1 By using (1.2) 1 X 1 A + A 1 1 X AA + By using (2.2) Therefore Y A 1 X A C and D CA + B 1 X AA + 1 X. Thus A M 1 X A 1 X. Theorem 2.3. Let A B F U M and L C D H K be range symmetric matrices in M both of the form (2.1) and ML of rank r, then the following are equivalent: (i) ML is range symmetric in M. (ii) AF is range symmetric in M and CA + HF + (iii) AF is range symmetric in M and A + B F + U Proof. Since M and L are of the form (2.1) by Lemma 2.1 there exist r (n r) matrices X and Y such that A F F Y M 1 X A 1 X and L 1 Y F 1 Y. F Y Now ML A 1 X A 1 X F 1 Y F F Y 1 Y F Y A(I X 1 Y )F A(I X 1 Y )F Y 1 X A(I X 1 Y )F 1 X (I X 1 Y )F Y AZF AZF Y 1 X AZF 1 X AZF Y where Z I X 1 Y. Clearly N(AZF ) N( 1 X AZF ) N( 1 X AZF ), N(AZF ) N(AZF Y ) and the Schur complement of AZF in ML is zero. For ML/AZF 1 X AZF Y + 1 X (AZF )(AZF ) + AZF Y 1 X AZF Y + 1 X AZF Y 0.

7 Product of Range Symmetric Block Matrices in Minkowski Space 65 Hence rk(azf ) rk(ml) r. Thus ML is also of the form (2.1). Since M and L are range symmetric in M by Theorem 2.2 and Lemma 2.1, A and F are range symmetric in M. Now 1 (A + ) 1 X (A + A) By Theorem X 1 (A + A) 1 By using X 1 A + A 1 1 X AA + By using 2.2 Similarly it can be proved that 1 Y F F + 1 (F + F Y ). We now claim AZF is range symmetric in M. N(F ) N(AZF ), and rk(azf ) rk(f ) r, hence it follows N(F ) N(AZF ). Also N(A ) N(AZF ) and rk(azf ) rk(azf ) rk(a) r rk(f ), N(A) N(A ) N(AZF ) N(F ). Thus N(AZF ) N(AZF ) and hence AZF is range symmetric in M. By Theorem 1.5 (ii), 1 AZF in EPr and by using (2.2) we have (2.4) 1 AZF (AZF ) + 1 (AZF ) + AZF By using (2.4) for we get (2.5) N(AZF ) N(F ), N(AZF ) N(A ) N(A) AZF (AZF ) + F F + 1 (AZF ) + AZF 1 and (AZF ) + AZF A + A 1 AA + 1. Since H 1 Y F and C 1 X A, we have Similarly by using (2.5), we have Therefore HF + 1 Y F F + by (2.5) 1 Y 1 AZF (AZF ) + 1 Y 1 (AZF ) + AZF 1 1 Y (AZF ) + AZF 1 (AZF ) + AZF Y. CA + 1 X AZF (AZF ) +. (2.6) CA + HF + 1 X AZF (AZF ) + 1 (AZF ) + AZF Y. Now the proof runs as follows: ML is range symmetric in M AZF is range symmetric in M and 1 X AZF (AZF ) + 1 (AZF ) + AZF Y AZF is range symmetric in M and CA + HF + By using (2.6) N(AZF ) N(AZF ) and CA + HF + N(F ) N(A) N(A) and CA + HF + AF is range symmetric in M and CA + HF + By using Theorem 2.1 AF is range symmetric in M and A + B F + U By Theorem 2.2.

8 66 AR. Meenakshi and D. Krishnaswamy 3. Factorization In this section a set of conditions under which a matrix can be expressed as product of range symmetric matrices in m are derived. Definition 3.1. A matrix A C n n is said to be unitary in unitary space if and only if AA A A I. Lemma , Theorem 1 Let A Cr n n be EPr matrix, then there exist a unitary matrix U such that A U DU U D1 0 U, where D 1 is r r non-singular. Lemma 3.2. Let A C n n, A is range symmetric in M if and only if A P D 0 P, where D is r r matrix and rk(d) r and P is unitary. Proof. By Theorem 1.5 (ii), the matrix A is range symmetric in M A is EP. By Lemma 3.1, this holds if and only if or or A U A U D1 0 D A U 0 I U U D1 0 U where 1 is r r or 1 D A U 1 0 U. Thus the matrix A is range symmetric in M if and only if A P D 0 P, where P U, P U U and D 1 D 1 is r r non-singular. Lemma 3.3. Let A and B be range symmetric matrices in M of rank r. Then N(A) N(B) if and only if N(P AP ) N(P BP ) where P is unitary.

9 Product of Range Symmetric Block Matrices in Minkowski Space 67 Proof. Let A and B be range symmetric matrices in M. Assume N(A) N(B). Then x N(P AP ) P AP x 0 AP x 0 Ay 0 where y P x y N(A) N(B)By 0 BP x 0 P BP x 0 x N(P BP ). Thus N(A) N(B) N(P AP ) N(P BP ). The proof the converse is similar and hence omitted. Theorem 3.1. Let M be of the form (2.1) be range symmetric in M. Then M can be written as a product of range symmetric matrices in M. Proof. Since M is of the form (2.1) and M is range symmetric in M, by Lemma 2.1, A is range symmetric in M and M A 1 X A 1 X. Since M is range symmetric in M by Theorem 1.5(ii), M in EPr, where M I A 1 X A 1 X 1 A 1 X A 1 1 X. Consider P 1 AA AA + 1 X X 1 AA + 1 X 1 AA + 1 X 1 A 0 L, 0 I A Q + A A + X A + A X A +., By using (1.1), P ( 1 AA + 1 ) (X 1 AA + 1 ) ( 1 AA + 1 X) (X 1 AA + 1 X) 1 (AA + ) 1 1 (AA + ) 1 X X 1 (AA + ) 1 X 1 (AA + ) 1 X 1 AA AA + 1 X X 1 AA + 1 X 1 AA + P 1 X Similarly Q Q can be proved. Thus P P and Q Q and therefore P, Q are EPr. Since A is range symmetric by Theorem 1.5(iii) 1 A is EPr and hence L is EPr.

10 68 AR. Meenakshi and D. Krishnaswamy Now P LQ 1 AA AA + 1 X 1 A 0 A + A X 1 AA + 1 X 1 AA + 1 X X A + A 1 AA A 0 A + A A + X 1 AA A 0 X A + A X A + 1 A 0 A + A A + X 1 A 0 X A + A X A + 1 AA + A 1 AA + X 1 AA + A X 1 AA + 1 A 1 X 1 A X 1 A 1 X A 1 X M. A + X A + By using (1.1), M P LQ (P )(L)(Q). Since P, Q, L and EPr, by Theorem 1.5 (ii) and (iii), it follow that P, L, Q are range symmetric M. Thus a range symmetric matrix M in M is expressed as a product of range symmetric matrices in M. Acknowledgement. The first author is thankful to the All India council for Technical Education, New Delhi for the financial support to carry out the work. References 1 T. S. Baskett and I. J. Katz, Theorems on products of EP r matrices, Linear Algebra and Appl. 2 (1969), A. Ben-Israel and T. N. E. reville, eneralized Inverses: Theory and Applications, Wiley- Intersci., New York, D. Carlson, E. Haynsworth and T. Markham, A generalization of the Schur complement by means of the Moore-Penrose inverse, SIAM J. Appl. Math. 26 (1974), A. R. Meenakshi, On Schur complements in an EP matrix, Period. Math. Hungar. 16 (1985), no. 3, A. R. Meenakshi, eneralized inverses of matrices in Minkowski space, in Proc. Nat. Seminar Alg. Appln. Annamalai University, Annamalainagar, (2000), A. R. Meenakshi, Range symmetric matrices in Minkowski space, Bull. Malays. Math. Sci. Soc. (2) 23(1) (2000), A. R. Meenakshi and D. Krishnaswamy, On sums of range symmetric matrices in Minkowski space, Bull. Malays. Math. Sci. Soc. (2) 25(2) (2002), M. Renardy, Singular value decomposition in Minkowski space, Linear Algebra Appl. 236 (1996), M. Pearl, On normal and EP r matrices, Michigan Math. J. 6 (1959), 1 5.