SOME PROPERTIES OF COMMON HERMITIAN SOLUTIONS OF MATRIX EQUATIONS $A_{1}XA_{1}^{*}=B_{1}$ AND $A_{2}XA_{2}^{*}=B_{2}$

Abstract

In this paper we provide necessary and sufficient conditions for the pair of matrix equations $ A_{1}XA_{1}^{*}=B_{1} $ and $ A_{2}XA_{2}^{*}=B_{2} $ to have a common hermitian solution in the form $ \frac{X_{1}{+}X_{2}}{2} $, where $ X_{1} $ and $ X_{2} $ are hermitian solutions of the equations $ A_{1}XA_{1}^{*}=B_{1} $ and $ A_{2}XA_{2}^{*}=B_{2}$ respectively.