On nding the generalized inverse matrix for the product of matrices

Transcription

1 PU.M.A. Vol , No. 3, pp. 997 On nding the generalized inverse matrix for the product of matrices X. M. Ren Department of Mathematics, Xi'an University of Architecture Technology, Xi'an, 70055, P.R.China. x.m.ren@@263.net Y. Wang Department of Mathematics, Xi'an University of Architecture Technology, Xi'an, 70055, P.R.China. K. P. Shum Faculty of Science, The Chinese University of Hong Kong, Hong Kong. kpshum@@math.cuhk.edu.hk Received: April, 2005 Abstract. Matrix computation is an important topic in applied mathematics information science. There are various methods of nding the inverse generalized inverse of a given matrix. However, for the product matrices, there does not exist a general method of nding its generalized inverse. In this note, we introduce the concept of swich sets of matrices. By using the new concept of swich sets, we are able to provide a method for nding a generalized inverse of product matrices. Mathematics Subject Classications A09 Generalized inverses of matrices its applications have been investigated by Rao Mitra in [3]. For some special matrices, some authors have given some interesting methods for nding their generalized inverse matrices. For example, Rakha [4] has recently given a method of nding the Moor-Penrose generalized inverse matrix. Furthermore, Werner [6] in 994 also described the problem for nding a generalized inverse for the product of matrices. In fact he considered the problem when will B A be a generalized inverse of AB? The matrix computation for information systems was also discussed by J. Guan, Bell Z. Guan in []. In this aspect, a recursive method for nding the inverse of a CSP matrix was rst provided by Ramabhadra Sharma in [2]. However, up to the present moment, except the paper by Werner [6], there does not exist a general method of nding a generalized inverse for the product of matrices. In this note, we will rst introduce the concept of swich sets of matrices. Then by using the swich sets of matrices, we will provide an eective method for This research is supported by Natural Science Foundation of Shaanxi Province 2004A0; The SF of Education Ministry of Shaanxi Province 05JK240, P.R.China. 2 This research is partially supported by a RGCCUHK direct grant # /

2 92 X. M. REN, Y. WANG AND K. P. SHUM nding a generalized inverse matrix for the product of matrices. Some examples will be demonstrated how to nd such a generalized inverse matrix for a product of some particular matrices. We rst denote the set of all m n matrices over a eld F by F n m. Let A F m n. If there exists some X F n m such that AXA = A XAX = X then we call this matrix X a reexive generalized inverse of the matrix A, denoted by A. For any A F m n, we can easily see that there exists A F n m. Now, we denote the rank of the matrix A by rank A = r. If r = 0, then A is an m n zero matrix, so the n m zero matrix O is a reexive generalized inverse matrix. If r 0, then there exists some invertible matrices P F m m Q F n n such that A = P [ Ir ] Q, 2 where P is the usual inverse matrix of the matrix P. In this case, we can verify that [ ] A Ir B = Q P 3 B 2 B 2 B for any B F r m r B 2 F n r r. From the matrix A with the form 3 above, we can see that for any no zero matrix A F m n, A is unique if only if A is invertible see [3]. Now, we denote the set of all the reexive generalized inverses A of a matrix A by V A. Clearly, the set V A is non-empty for any matrix A. Let E, be the set of all n n idempotent matrices, that is, E = {E : E 2 = E, E F n n }. Then, we can easily see that for any A F m n, AA A A are both idempotent matrices. In order to obtain a reexive generalized inverse for product matrices, we now introduce the following denition. Definition Suppose that E, F F n n E. Then we call SE, F = {G E : GE = F G = G EGF = EF } the swich set of the matrices E F. The swich sets have the following properties. Proposition 2 i SE, F dened above is non-empty.

3 ON FINDING THE GENERALIZED INVERSE MATRIX 93 ii SE, F = if only if GH = HG for any G, H SE, F. iii For any E E, SE, E contains a unique idempotent matrix E, i.e. SE, E = {E}. iv Suppose that I is the usual identity matrix. Then SI, I contains a unique identity matrix I. Proof. i It is clear that for any idempotent matrices E, F F n n, its product EF is also an n n matrix. Let P V EF, G = F P E. Then, we have G 2 = F P E F P E = F P EF P E = F P E = G so that G is an idempotent matrix. Also, by denition formula, we can see that GE = F P E E = F P E = G F G = F F P E = F P E = G EGF = EF P EF = EF P EF = EF. This shows that G SE, F hence the proof is completed. ii The necessity is immediate since every matrix G in SE, F is an idempotent matrix. We now prove the suciency. Suppose that G, H SE, F. Then, by denition of the swich set, it is evident that G, H E such that This leads to GE = F G = G EHF = EF. GHG = GEHF G = GEHF G = GEF G = G 2 = G. By a similar argument, we can also deduce that HGH = H. Thus, by our hypothesis, it follows that This shows that SE, F =. G = GHG = G 2 H = GH = GH 2 = HGH = H. iii Suppose that G SE, E. Then, by denition of the swich set, we have E G E = E 2 = E EG = GE = G. Hence, E = EGE = GE = G. iv Part iv follows immediately from iii. We are now ready to provide a method of nding a reexive generalized inverse for the product of some particular matrices. We give the following theorem. Theorem 3 Suppose that A F m n B F n p such that A V A B V B. Then B GA V AB for any G SA A, BB.

4 94 X. M. REN, Y. WANG AND K. P. SHUM Proof. It is easy to see that A A BB are both n n idempotent matrices. Now, we write A A = E BB = F. Then by using the denition of swich set of the matrices E F, for any G SE, F, we have ABB GA AB = ABB GA AB On the other h, we also have = AF GEB = AGB = AA AGBB B = AEGF B = AEF B = AA ABB B = AB. B GA ABB GA = B GEF GA = B G 2 A = B GA. Hence, by the denition of the reexive generalized inverses matrix, we can see immediately that B GA V AB. The following corollaries are consequences of Theorem 3 Proposition 2 ii. Corollary 4 Suppose that A F m n B F n p. If A V A B V B such that A A = BB = E E, then B A V AB. Corollary 5 Suppose that A F m n B F n p. If A V A B V B such that A A = BB = I, then B A is a reexive generalized inverse matrix for the product AB of the matrices A B. Corollary 6 i If A is an n n invertible matrix with the inverse A B is an n p matrix, then for any B V B, the product B A V AB. ii If A is an m n matrix B is an n n invertible matrix with the inverse matrix B, then for any A V A, the product B A V AB. Proof. We only need to prove part i because the proof of part ii is similar. By our hypothesis, we see that an n n matrix A is invertible so A A is clearly the identity matrix I. Hence, we only need to consider the swich set SI, BB, for any B V B. In this cases, it can be veried that the idempotent matrix BB is in SI, BB. Consequence, by Theorem 3, we immediately see that B A V AB. Thus, the proof is completed. Corollary 7 Suppose that A F m B F p. Then for any A V A B V B, the product matrix B A is a reexive generalize inverse matrix for the product AB of the matrices A B. Proof. The conclusion is obvious because the swich set SA A, BB contains a unique element. We now give some examples below to demonstrate how to apply our theorem to nd the generalized inverse matrix for some product of particular matrices.

5 ON FINDING THE GENERALIZED INVERSE MATRIX 95 Example 8 Let A = a, a 2,..., a m T B = b, b 2,..., b p, where a, b 0. We now nd a reexive generalized inverse matrix for AB. According to our formula 3, we can easily see that where A = + CA, c, c 2,..., c m, a A = a 2,..., a T m a a C = c, c 2,..., c m, which is an arbitrary m matrix. Similarly, we have T B = + B D, d..., d p, b where B = b 2,..., b p b b D = d,..., d p T which is an arbitrary p matrix. By using our Corollary 4, we have + B D b AB = B A = d + CA, c, c 2,..., c m.. a d p In particular, if we take A = B =, 0, 0,..., 0 a T, 0,..., 0, b then we immediately obtain a generalized inverse of AB as follows 0 0 a AB b =

6 96 X. M. REN, Y. WANG AND K. P. SHUM Example 9 Suppose that A = B = T In order to nd a reexive generalized inverse matrix for the product of matrices AB, we rst nd the set V A by using our formula 3. In fact, we can easily verify that { 3 + a 3b 2 + a b a b V A = a, b, c, d F} 2 + c 3d + c 3d c d where V B =. { } B B B B = 2 B 3 B 4 B 5, B 2 B 22 B 23 B 24 B 25 B = + e g + f h B 2 = e f i j B 3 = e f B 4 = g h B 5 = j i B 2 = 2 + 2e 2g + f h B 22 = 3 2e f 2i j B 23 = 2e f B 24 = 2g h B 25 = 2i j for e, f, g, h, i, j F. Now, we can nd a reexive generalized inverse matrix for the product of the matrices A B, that is, the matrix AB. If we choose A = then we have A A = 0 B = BB = By Corollary 5, we immediately obtain that AB 0 0 = B A = 0 3.

7 ON FINDING THE GENERALIZED INVERSE MATRIX 97 References = [] J.W. Guan, D.A. Bell Z. Guan, Matrix computation for information systems, Information Sciences, , [2] I. Ramabhadra Sharma, A.V. Dattatreya Rao B. Rami Reddy, A recursive method for nding the inverse of a CSP matrix, Information Sciences, , [3] C.R. Rao S.K. Mitra, Generalized Inverse of Matrices Its Application, Wiley, New York, 97. [4] M.A. Rakha, On the Moore-Penrose generalized inverse matrix, Applied Mathematics Computation, , [5] B. Zheng R.B. Bapat, Generalized inverse A 2 T,S a rank equation, Applied Mathematics Computation, , [6] H.J. Werner, When is B A a generalized inverse of AB? Linear Algebra its Applications, 2994,