Bisymmetric and Centrosymmetric Solutions to Systems of Real Quaternion Matrix Equations

A1X = C1, A1X = C1, XB3 = C3, A2X = 62, to have bisymmetric solutions, and the system A1X = Ca, A3X B3 = C3, to have centrosymmetric solutions. The expressions of such solutions of the matrix and the systems mentioned above are also given. Moreover a criterion for a quaternion matrix to be bisymmetric is established and some auxiliary results on other sets over H are also mentioned. ~) 2005 Elsevier Ltd. All rights reserved. K e y w o r d s S y s t e m of quaternion matrix equations, Inner inverse of a matrix, Reflexive inverse of a matrix, Centrosymmetric matrix, Bisymmetric matrix. 1. I N T R O D U C T I O N In [1], Khatri and Mitra studied the Hermitian solutions to the following matrix equations over the complex field: A X = C, (1.1) A X B = C, (1.2) This research was supported by the Natural Science Foundation of China (0471085), the Natural Science Foundation of Shanghai, the Development Foundation of Shanghai Educational Committee (214498), and the Special Funds for Major Specialities of Shanghai Education Committee. The author is very grateful to the referees for their useful comments and suggestions. 0898-1221/05/$ see front matter (~ 2005 Elsevier Ltd. All rights reserved. Typeset by .AA/tS-TEX doi:10.1016/j.camwa.2005.01.014 642 Q.-W. WANG and A 1 X = C1, X B 3 = C3. (1.3) Vetter [2], Magnus and Neudecker [31, Don [4], Dai [5], Navarra, Odell and Young [6], and others studied the symmetric solution, Hermitian solutions to the matrix equation (1.2). Centrosymmetric and bisymmetric matrices have been widely discussed since 1939, which are very useful in engineering problems, information theory, linear system theory, linear estimation theory and numerical analysis theory, and others (e.g., [7-17]). So investigating centrosymmetric solutions and bisymmetric solutions of matrix equations should be significant and interesting. Inspired by Navarra, Odell and Young [6], in order to investigate centrosymmetric solutions and bisymmetric solutions to some matrix equations, we in [18] considered the system of matrix equations A 1 X = C1, A 2 X = C2, A 3 X B 3 = C3, A 4 X B 4 : C4, (1.4) over the real quaternion algebra ]E = {ao + a l i + a2j ÷ a3k I i2 = j2 = k 2 = i j k = 1 and ao, a l ,a2 ,a3 E •}, where 1~ is the real number field. A necessary and sufficient condition for the existence and an expression of the general solution to system (1.4) were derived. As a special case of the system, the following system A 1 X = C1, A ~ X = C2, X B3 = C3, X B 4 = C4, (1.5) derive a necessary and sufficient condition for the existence of the centrosymmetric solution and its representation to the system A 1 X = C1, A a X B a = C3, (1.7) over H. Throughout we denote the set of all m x n matrices over H by ]HI m×~, the identity matrix with the appropriate size by I , an inner inverse of a matrix A over H by AO) which satisfies AA(1)A -A, a reflexive inverse of a matrix A over ]HI by A + which satisfies simultaneously A A + A = A and A + A A + = A +. Moreover, LA =I A+A, RA : I -A A + where A + is an any but fixed reflexive inverse of A. Clearly, LA and RA are idempotent and one of its reflexive inverses, respectively. The following results of [18] will be used in the sequel. was also considered. In this paper, we use the results of [18] to consider bisymmetric solutions and centrosymmetric solutions to some matrix equations over H. In Section 2, we first derive a criterion for a bisymmetric matrix over H, then give necessary and sufficient conditions for the existence and the expressions of bisymmetric solutions to the matrix equation (1.1) and system (1.3) and A 1 X = C1, A 2 X = C2, (1.6) Bisymmetric and Centrosymmetric Solutions 643 LEMMA 1.1. (See Theorem 2.3 in [18].) Suppose that A1 ~ ~'~×~, C~ ~ ~m×~, A2 e ~ × ~ , C~ ~ H~x~, A3 ~ ~kx~, B3 ~ [4 ~xp, C3 ~ ~ x v , Aa ~ ~ x ~ , B4 ~ ~rx~, C4 ~ ~qxl are known and X ~ ~ × ~ unknown; and S = A2LAI, K = A3LA~, T = K L s , G = RsA2, M = A4LA~, N = flB3B4, P -RMLsLTMLs , ~J = A3 [A+ C3B + A+ C1 LA~S+ A2 (A+C2 A+C1) ] B3, = S+A2 (A+C2 A+Cl) + LsT+~B+3, Q = C4 A4A+C1B4 Mq~B4, then system (1.4) is consistent if and only if T T + ~ = ~, RpRMLsLTQ = O, RMLsLTQLN = 0,