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Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations. (English) Zbl 1138.15003
Summary: In this paper we consider bisymmetric and centrosymmetric solutions to certain matrix equations over the real quaternion algebra $\Bbb H$. Necessary and sufficient conditions are obtained for the matrix equation $AX = C$ and the following systems $$\aligned A_1X&=C_1,\\ XB_3&=C_3,\endaligned \quad \quad \aligned A_1X&=C_1,\\ A_2X&=C_2,\endaligned$$ to have bisymmetric solutions, and the system $$\align A_1X & =C_1,\\ A_3 XB_3 & =C_3,\endalign$$ 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 $\Bbb H$ are also mentioned.

MSC:
15A24Matrix equations and identities
15B33Matrices over special rings (quaternions, finite fields, etc.)
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References:
[1] Khatri, C. G.; Mitra, S. K.: Hermitian and nonnegative definite solutions of linear matrix equations. SIAM J. Appl. math. 31, 578-585 (1976) · Zbl 0359.65033
[2] Vetter, W. J.: Vector structures and solutions of linear matrix equations. Linear algebra appl. 9, 181-188 (1975) · Zbl 0307.15003
[3] Magnus, J. R.; Neudecker, H.: The elimination matrix: some lemmas and applications. SIAM J. Algebraic discrete methods 1, 422-428 (1980) · Zbl 0497.15014
[4] Don, F. J. Henk: On the symmetric solutions of a linear matrix equation. Linear algebra appl. 93, 1-7 (1987) · Zbl 0622.15001
[5] Dai, H.: On the symmetric solution of linear matrix equations. Linear algebra appl. 131, 1-7 (1990) · Zbl 0712.15009
[6] Navarra, A.; Odell, P. L.; Young, D. M.: A representation of the general common solution to the matrix equations A1XB1 = C1 and A2XB2 = C2 with applications. Comput. math. Applic. 41, No. 7/8, 929-935 (2001) · Zbl 0983.15016
[7] Aitken, A. C.: Determinants and matrices. (1939) · Zbl 0022.10005
[8] Datta, L.; Morgera, S. D.: On the reducibility of centrosymmetric matrices-applications in engineering problems. Circuits systems sig. Proc. 8, 71-96 (1989) · Zbl 0674.15005
[9] Cantoni, A.; Butler, P.: Eigenvalues and eigenvectors of symmetric centrosymmetric matrices. Linear algebra appl. 13, 275-288 (1976) · Zbl 0326.15007
[10] Weaver, J. R.: Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, eigenvectors. Amer. math. Monthly 92, 711-717 (1985) · Zbl 0619.15021
[11] Lee, A.: Centrohermitian and skew-centrohermitian matrices. Linear algebra appl. 29, 205-210 (1980) · Zbl 0435.15019
[12] Hell, R. D.; Bates, R. G.; Waters, S. R.: On centrohermitian matrices. SIAM J. Matrix anal. Appl. 11, No. 1, 128-133 (1990) · Zbl 0709.15021
[13] Hell, R. D.; Bates, R. G.; Waters, S. R.: On perhermitian matrices. SIAM J. Matrix anal. Appl. 11, No. 2, 173-179 (1990) · Zbl 0709.15022
[14] Reid, R. M.: Some eigenvalues properties of persymmetric matrices. SIAM rev. 39, 313-316 (1997) · Zbl 0876.15006
[15] Andrew, A. L.: Centrosymmetric matrices. SIAM rev. 40, 697-698 (1998) · Zbl 0918.15006
[16] Pressman, I. S.: Matrices with multiple symmetry properties: applications of centrohermitian and perhermitian matrices. Linear algebra appl. 284, 239-258 (1998) · Zbl 0957.15019
[17] Melman, A.: Symmetric centrosymmetric matrix--vector multiplication. Linear algebra appl. 320, 193-198 (2000) · Zbl 0971.65022
[18] Wang, Q. W.: The general solution to a system of real quaternion matrix equations. Comput. math. Applic. 49, No. 5/6, 665-675 (2005) · Zbl 1138.15004
[19] Wang, Q. W.; Sun, J. H.; Li, S. Z.: Consistency for $bi(skew)$symmetric solutions to systems of generalized Sylvester equations over a finite central algebra. Linear algebra appl. 353, 169-182 (2002) · Zbl 1004.15017
[20] Wang, Q. W.: A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity. Linear algebra appl. 384, 43-54 (2004) · Zbl 1058.15015