The general common Hermitian Nonnegative-definite solution to the matrix equations AXA*=B and CXC*=D

Abstract

We give a necessary and sufficient condition for the existence of at least one common Hermitian nonnegative-definite (positive-definite) solution to the pair of matrix equations AXA*=B and CXC*=D, and derive a representation of the general common Hermitian nonnegative-definite (positive-definite) solution to these two equations when they have at least such one common solution. This article can be viewed as a supplementary version of that derived by Young et al. (D.M. Young, J.W. Seaman Jr. and L.M. Meaux (1999). Independence distribution preserving covariance structures for themultivariate model. J. Multivariate Anal., 68, 165–175) since Groß (J. Groß (2000). Nonnegative-define and positive solutions to the matrix equation AXA*=B – revisited. Linear Algebra Appl., 321, 123–129) has given a counterexample to point out a mistake in their basic Theorem 1. The proposed approach is convenient to use and possesses good numerical reliability since it mainly involves only a series of singular value decompositions and the inverses of several positive definite diagonal matrices.