Research Article Completing a 2 2Block Matrix of Real Quaternions with a Partial Specified Inverse

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1 Applied Mathematics Volume 0, Article ID 7978, 5 pages Research Article Completing a Block Matrix of Real Quaternions with a Partial Specified Inverse Yong Lin, and Qing-Wen Wang Department of Mathematics, Shanghai University, Shanghai 00444, China School of Mathematics and Statistics, Suzhou University, Suzhou 4000, China Correspondence should be addressed to Qing-Wen Wang; wqw858@yahoo.com.cn Received 4 December 0; Revised February 0; Accepted 0 March 0 Academic Editor: K. Sivakumar Copyright 0 Y. Lin and Q.-W. Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper considers a completion problem of a nonsingular block matrix over the real quaternion algebra H: Let m, m, n, n be nonnegative integers, m +m =n +n =n>0,anda H m n,a H m n,a H m n,b H n m be given. We determine necessary and sufficient conditions so that there exists a variant block entry matrix A H m n such that A=( A A A A ) H n n is nonsingular, and B is the upper left block of a partitioning of A. The general expression for A is also obtained. Finally, a numerical example is presented to verify the theoretical findings.. Introduction The problem of completing a block-partitioned matrix of a specified type with some of its blocks given has been studied by many authors. Fiedler and Markham [] considered the following completion problem over the real number field R. Suppose m, m, n, n are nonnegative integers, m +m = n +n =n>0, A R m n, A R m n, A R m n, and B R n m. Determine a matrix A R m n such that A=( A A A A ) () is nonsingular and B is the lower right block of a partitioning of A.Thisproblemhastheformof ( A A A? ) = (?? ), ()? B and the solution and the expression for A were obtained in [].Dai [] considered this form of completion problems with symmetric and symmetric positive definite matrices over R. Some other particularforms for block matrices over R have also been examined (see, e.g., []), such as ( A A A? ) = ( B??? ), ( A??? ) = (??? B ), ( A? ) = (? B? A B? ). The real quaternion matrices play a role in computer science, quantum physics, and so on (e.g., [4 6]). Quaternion matrices are receiving much attention as witnessed recently (e.g., [7 9]). Motivated by the work of [, 0] andkeeping such applications of quaternion matrices in view, in this paper we consider the following completion problem over the real quaternion algebra: H = {a 0 +a i+a j+a k i =j =k =ijk= and a 0,a,a,a R}. Problem. Suppose m, m, n, n are nonnegative integers, m +m = n +n = n > 0,andA H m n, () (4)

2 Applied Mathematics A H m n, A R m n, B H n m. Find a matrix A H m n such that A=( A A A A ) H n n (5) is nonsingular, and B is the upper left block of a partitioning of A.Thatis (? A ) A A =( B? ), (6)?? where H m n denotes the set of all m nmatrices over H and A denotes the inverse matrix of A. Throughout, over the real quaternion algebra H, we denote the identity matrix with the appropriate size by I,the transpose of A by A T,therankofA by r(a), theconjugate transpose of A by A =(A) T, a reflexive inverse of a matrix A over H by A + which satisfies simultaneously AA + A=Aand A + AA + =A +.Moreover,L A = I A + A, R A = I AA +,where A + is an arbitrary but fixed reflexive inverse of A.Clearly,L A and R A are idempotent, and each is a reflexive inverse of itself. R(A) denotes the right column space of the matrix A. The rest of this paper is organized as follows. In Section, we establish some necessary and sufficient conditions to solve Problem over H, and the general expression for A is also obtained. In Section, we present a numerical example to illustrate the developed theory.. Main Results In this section, we begin with the following lemmas. Lemma (singular-value decomposition [9]). Let A H m n be of rank r. Then there exist unitary quaternion matrices U H m m and V H n n such that UAV=( D r 0 ), (7) 0 0 where D r = diag(d,...,d r ) and the d j s are the positive singular values of A. Let H n c denote the collection of column vectors with n components of quaternions and A be an m nquaternion matrix. Then the solutions of Ax=0formasubspace of H n c of dimension n(a).wehavethefollowinglemma. Lemma. Let ( A A A A ) (8) be a partitioning of a nonsingular matrix A H n n,andlet ( B B B B ) (9) be the corresponding (i.e., transpose) partitioning of A.Then n(a )=n(b ). Proof. It is readily seen that ( B B B B ), ( A A A A ) (0) are inverse to each other, so we may suppose that n(a )< n(b ). If n(b )=0,necessarilyn(A )=0and we are finished. Let n(b )=c>0, then there exists a matrix F with c right linearly independent columns, such that B F = 0.Then, using we have From we have A B +A B =0, () A B F=0. () A B +A B =I, () A B F=F. (4) It follows that the rank r(b F) c. Inviewof(), this implies Thus n (A ) r(b F) c=n(b ). (5) n(a )=n(b ). (6) Lemma (see [0]). Let A H m n, B H p q, D H m q be known and X H n p unknown. Then the matrix equation is consistent if and only if In that case, the general solution is AXB = D (7) AA + DB + B=D. (8) X=A + DB + +L A Y +Y R B, (9) where Y, Y are any matrices with compatible dimensions over H. By Lemma, letthesingularvaluedecompositionofthe matrix A and B in Problem be A =Q( Λ )R, (0) B =U( Σ )V, ()

3 Applied Mathematics where Λ=diag(λ,λ,...,λ s ) is a positive diagonal matrix, λ i =0 (i =,...,s) are the singular values of A, s = r(a ), Σ=diag(σ,σ,...,σ r ) is a positive diagonal matrix, σ i =0 (i =,...,r) are the singular values of B and r = r(b ). Q = (Q Q ) H m m, R = (R R ) H n n, U=(U U ) H n n, V=(V V ) H m m are unitary quaternion matrices, where Q H m s, R H n s, U H n r,andv H m r. Theorem 4. Problem has a solution if and only if the following conditions are satisfied: (a) r( A A )=n, (b) n r(a )=m r(b ),thatisn s=m r, (c) R(A B ) R(A ), (d) R(A B ) R(A ). In that case, the general solution has the form of A =B + +A R( Λ Q A U Σ 0 H (V A R ) ) V B + +Y YB B +, () (n where H is an arbitrary matrix in H s) r and arbitrary matrix in H m n. Y is an Proof. If there exists an m n matrix A such that A is nonsingular and B is the corresponding block of A,then (a) is satisfied. From AB = BA = I,wehavethat A B +A B =0, B A +B A =0, so that (c) and (d) are satisfied. By (), we have () r(a )+n(a )=n, r(b )+n(b )=m. (4) From Lemma, Noticethat( A A A A ) is the corresponding partitioning of B,wehave n (B ) =n(a ), (5) implying that (b) is satisfied. Conversely, from (c), we know that there exists a matrix K H n m such that Let From (0), (), and (6), we have A B =A K. (6) B = K. (7) A U( Σ )V =Q( Λ )R K. (8) It follows that Q A U( Σ )V V=Q Q( Λ )R KV. (9) This implies that ( Q A U Q A U Q A U Q A U )( Σ ) =( Λ )(R KV R KV R KV R KV ). Comparing corresponding blocks in (0), we obtain (0) Q A U =0. () Let R KV = K.From(9), (0), we have K =( Λ Q A U Σ 0 H K ), H H (n s) r, K H (n s) (m r). Inthesameway,from(d),wecanobtain () V A R =0. () Notice that ( A A ) in (a) is a full column rank matrix. By (0), (), and (), we have Λ 0 ( 0 Q V 0 )(A 0 0 )R=( A V A R V A ), (4) R V A R V A R so that n =r( A )=r(( 0 Q A V 0 )(A )R) A Λ =r( V A R V A ) R V A R V A R =r(λ) +r(v A R ) =s+r(v A R ). (5) It follows from (b) and (5) thatv T A R is a full column rank matrix, so it is nonsingular. From AB = I, we have the following matrix equation: that is A B +A B =I, (6) A B =I A B, I H m m, (7)

4 4 Applied Mathematics where B, A were given, B = K (from (7)). By Lemma, the matrix equation (7)hasasolutionifandonly if (I A B )B + B =I A B. (8) By (), (7), (), and (), we have that (8)isequivalentto:. An Example In this section, we give a numerical example to illustrate the theoretical results. Example 5. Consider Problem with the parameter matrices as follows: (I + A K) V ( Σ )U U( Σ )V =I+A K. (9) We simplify the equation above. The left hand side reduces to (I + A K)V V and so we have A KV V A K =I V V. (40) So, A R KV V V A R KV =(V V )( V V ) V V. (4) This implies that so that So, and hence, A R K( V V V V )V A R K( V V )=V V, (4) A R K( I 0 )V A R K( V V )=V V. (4) A R K( 0 V )=V V, (44) (A R A R )( Λ Q A U Σ 0 )( 0 H K V )=V V. (45) Finally, we obtain A R K V = V V. (46) Multiplying both sides of (46)byV from the left, considering () and the fact that V A R is nonsingular, we have K = (V A R ). (47) From Lemma, (8), (47), Problem has a solution and the general solution is A =B + +A R( Λ Q A U Σ 0 H (V A R ) ) V B + +Y YB B +, (48) where H is an arbitrary matrix in H (n s) r and Y is an arbitrary matrix in H m n. +j k A =( k + ), j + i j k A =( j+ k + ), i A =( i j k ), B =( i j k ). It is easy to show that (c), (d) are satisfied, and that n =r( A A )=, n r(a )=m r(b )=0, so (a), (b) are satisfied too. Therefore, we have where B + =( j ), i k A =Q( Λ )R, B =U( Σ )V, We also have Q= i ( j k ), Λ=( 0 0 ), R=( 0 i ), U= ( 0 j k ), Σ=( 0 0 ), V=( 0 0 ). Q = i ( j k ), R =( 0 0 ), U = i ( j k ), V =( 0 0 ). (49) (50) (5) (5) (5)

5 Applied Mathematics 5 By Theorem 4, for an arbitrary matrices Y H,we have A =B + +A R(Λ Q A U Σ) V B + +Y YB B j+ 4 k i j =( i+ 4 j 4 k 4 ), 4 i j k it follows that + 4 j+ 4 k i j +j k A= ( i+ 4 j 4 k 4 4 i j k k + j + i j ), k i ( j+ k + i j k ) A =( 0 i j k j ). (54) i i j k (55) The results verify the theoretical findings of Theorem 4. [6] J. W. Shuai, The quaternion NN model: the identification of colour images, Acta Comput. Sinica, vol. 8, no. 5, pp. 7 79, 995 (Chinese). [7] F. Zhang, On numerical range of normal matrices of quaternions, Mathematical and Physical Sciences, vol. 9, no. 6, pp. 5 5, 995. [8]F.Z.Zhang,Permanent Inequalities and Quaternion Matrices [Ph.D. thesis], University of California, Santa Barbara, Calif, USA, 99. [9] F. Zhang, Quaternions and matrices of quaternions, Linear Algebra and Its Applications,vol.5,pp. 57,997. [0] Q.-W. Wang, The general solution to a system of real quaternion matrix equations, Computers & Mathematics with Applications,vol.49,no.5-6,pp ,005. Acknowledgments The authors would like to give many thanks to the referees and Professor K. C. Sivakumar for their valuable suggestions andcomments,whichresultedinagreatimprovementof the paper. This research was supported by Grants from the Key Project of Scientific Research Innovation Foundation of Shanghai Municipal Education Commission (ZZ080), the National Natural Science Foundation of China (705), the Natural Science Foundation of Shanghai (ZR4500), the Discipline Project at the corresponding level of Shanghai (A ), and Shanghai Leading Academic Discipline Project (J500). References [] M. Fiedler and T. L. Markham, Completing a matrix when certain entries of its inverse are specified, Linear Algebra and Its Applications, vol. 74, pp. 5 7, 986. [] H. Dai, Completing a symmetric block matrix and its inverse, Linear Algebra and Its Applications, vol. 5, pp. 5 45, 996. []W.W.Barrett,M.E.Lundquist,C.R.Johnson,andH.J. Woerdeman, Completing a block diagonal matrix with a partially prescribed inverse, Linear Algebra and Its Applications, vol. /4, pp. 7 87, 995. [4]S.L.Adler,Quaternionic Quantum Mechanics and Quantum Fields,OxfordUniversityPress,NewYork,NY,USA,994. [5] A. Razon and L. P. Horwitz, Uniqueness of the scalar product in the tensor product of quaternion Hilbert modules, Journal of Mathematical Physics,vol.,no.9,pp ,99.

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