The Re-nonnegative definite and Re-positive definite solutions of the matrix equation AXB = C on a linear manifold

Abstract

Let SE = { X ∈ Cn×n | ∥ XD - Z ∥F = min, D, Z ∈ Cn×q } be a linear manifold of Cn×n. We will consider the Re-nonnegative definite (Re-nnd) and Re-positive definite (Re-pd) solutions of the matrix equation AXB = C on SE , and deduce the solvability conditions for the equation as well as the explicit expressions for the general Re-nnd and Re-pd solutions when the stated conditions hold. MSC Classification: 15A24 , 15A57