Least Squares Solution of BXAT=T over Symmetric, Skew-Symmetric, and Positive Semidefinite X

Abstract

An efficient method based on the quotient singular value decomposition (QSVD) is used to solve the constrained least squares problem $\min \| T-BXA^T\|_F$ over symmetric, skew-symmetric, and positive semidefinite (maybe asymmetrical) X. The general expression of the solution is given and some necessary and sufficient conditions are derived about the solvability of the matrix equation BXAT=T. In each case, an algorithm is given for the unique solution when B and A are of full column rank.