Common Solutions to the Matrix Equations AX=B and XC=D on a Subspace.

Let S R Ω n × n be the set of all n × n symmetric matrices on subspace Ω , where Ω = { z ∈ R n | G z = 0 , G ∈ R k × n }. The necessary and sufficient conditions for the matrix equations A X = B and X C = D to have a common solution in S R Ω n × n and also an expression for the general common solution are obtained. Further, the associated optimal approximate problem to a given matrix X ~ ∈ R n × n is discussed and the optimal approximate solution is elucidated. Finally, a numerical experiment is presented to validate the accuracy of our result. [ABSTRACT FROM AUTHOR]