The rank-constrained Hermitian nonnegative-definite and positive-definite solutions to the matrix equation \(AXA^{\ast}=B\)
Given the matrices \(A\in {\mathbb C}^{m\times n}\) and \(B\in {\mathbb C}^{n\times n}\) (\(B\) being Hermitian nonnegative-definite (HNND)) which satisfy the conditions \(A\neq 0\), \(R(B)\subseteq R(A)\) (\(R(A)\) is the column space of \(A\)), and given the integer \(p\), give the necessary and sufficient conditions for the solvability of the equation \((*)\) \(AXA^*=B\) where \(X\) must be \(n\times n\), HNND and of rank \(p\). Next, under these conditions, determine the general HNND solution to \((*)\). The authors solve this problem by proposing an approach which possesses a good numerical reliability because it involves only two singular value decompositions and inverses of two positive-definite diagonal matrices. They consider also Hermitian positive-definite solutions. The approach is illustrated by an example.
scientific article; zbMATH DE number 1970616