The rank-constrained Hermitian nonnegative-definite and positive-definite solutions to the matrix equation \(AXA^{\ast}=B\)
type: Item
instance of: scholarly article
title: The rank-constrained Hermitian nonnegative-definite and positive-definite solutions to the matrix equation \(AXA^{\ast}=B\)
zbMATH Open document ID: 1026.15011
DOI: 10.1016/S0024-3795(03)00385-9
author: Xian Zhang
author: Mei-yu Cheng
published in: Linear Algebra and its Applications
publication date: 2003-08-25T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 1970616
zbMATH Keywords: rank-constrained solution
zbMATH Keywords: nonnegative-definite solution
zbMATH Keywords: numerical reliability
zbMATH Keywords: singular value decomposition
zbMATH Keywords: positive-definite solution
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1970616