Symmetric and skew-antisymmetric solutions to systems of real quaternion matrix equations. (English) Zbl 1143.15012

Li, Yao-Tang; Wu, Wen-Jing

Comput. Math. Appl. 55, No. 6, 1142-1147 (2008).

Summary: We consider symmetric and skew-antisymmetric solutions to certain matrix equations over the real quaternion algebra \(H\). First, a criterion for a quaternion matrix to be symmetric and skew-antisymmetric is given. Then, necessary and sufficient conditions are obtained for the matrix equation \(AX=C\) and the following system
\[ A_1X=C_1, \qquad XB_3=C_3 \]
to have symmetric and skew-antisymmetric solutions. The expressions of such solutions of the matrix equation and the system mentioned above are also given.

MSC:

15A24	Matrix equations and identities
15B33	Matrices over special rings (quaternions, finite fields, etc.)

Keywords:

real quaternion algebra; system of quaternion matrix equations; inner inverse; reflexive inverse; symmetric and skew-antisymmetric solutions

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. Algebr. Discrete Methods, 1, 422-428 (1980) · Zbl 0497.15014
[4]	Henk Don, F. J., 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 equation, Linear Algebra Appl., 131, 1-7 (1990)
[6]	Navarra, A.; Odell, P. L.; Young, D. M., A representation of the general common solution to the matrix equations \(A_1 X B_1 = C_1\) and \(A_2 X B_2 = C_2\) with applications, Comput. Math. Appl., 41, 7-8, 929-935 (2001) · Zbl 0983.15016
[7]	Wang, Q. W., Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations, Comput. Math. Appl., 49, 641-650 (2005) · Zbl 1138.15003
[8]	Wang, Q. W., The general solution to a system of real quaternion matrix equations, Comput. Math. Appl., 49, 5-6, 665-675 (2005) · Zbl 1138.15004
[9]	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