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

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

${A}_{1}X={C}_{1},\phantom{\rule{2.em}{0ex}}X{B}_{3}={C}_{3}$

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 15A33 Matrices over special rings