Symmetric Positive Semi - Definite Solutions of Bax = and Dxc =

In this paper, a sufficient and necessary condition for the matrix equations B AX = and , D XC = where , , n m n m B A × × ∈ ∈ R R , p n C × ∈R and , p n D × ∈ R to have a common symmetric positive semi-definite solution X is established, and if it exists, a representation of the solution set X S is given. An optimal approximation between a given matrix n n X × ∈ R ~ and the affine subspace X S is discussed, an explicit formula for the unique optimal approximation solution is presented, and a numerical example is provided.