On the Hermitian R -conjugate solution of a system of matrix equations

Let r be an n by n nontrivial real symmetric involution matrix,that is,r=r-1=rt≠in. an n×n complex matrix a is termed r-conjugate if ā=rar,where ā denotes the conjugate of a. we give necessary and sufficient conditions for the existence of the hermitian r-conjugate solution to the system of complex matrix equations ax=c and xb=d and present an expression of the hermitian r-conjugate solution to this system when the solvability conditions are satisfied. in addition,the solution to an optimal approximation problem is obtained. furthermore,the least squares hermitian r-conjugate solution with the least norm to this system mentioned above is considered. the representation of such solution is also derived. finally,an algorithm and numerical examples are given. © 2012 chang-zhou dong et al.