Unitary equivalence in an indefinite scalar product: An analogue of singular-value decomposition
type: Item
instance of: scholarly article
title: Unitary equivalence in an indefinite scalar product: An analogue of singular-value decomposition
zbMATH Open document ID: 0828.15009
DOI: 10.1016/0024-3795(93)00295-B
author: Yuri Bolshakov
author: Boris Raykhshteyn
published in: Linear Algebra and its Applications
publication date: 1995-09-06T00:00:00Z
review text: If \(H_1\) is an \(n\times m\) and \(H_2\) an \(m\times n\) invertible Hermitian matrix and \(X\) and \(Y\) are arbitrary complex \(m\times n\) matrices, the last matrices are called equivalent if \(X= U_2 YU_1\) for some \(H_1\)-unitary matrix \(U_1\) and some \(H_2\)-unitary matrix \(U_2\). If \(H_1\) and \(H_2\) are positive definite, then they are identities, and \(X\) and \(Y\) are equivalent if and only if the matrices \(X^* X\) and \(Y^* Y\) have the same spectrum. The authors show that the Jordan form of \(X^{[*]} X\), where \(X^{[*]}\) is the \(H_1\)-\(H_2\)-adjoint of \(X\), \(X^{[*]}= H^{- 1}_1 X^* H_2\), defines a finite number of nonequivalent classes of matrices and that each such class is defined by its integer matrix. Explicit formulas for all classes having the same Jordan form of \(X^* X\) are presented. The necessary and sufficient conditions for an operator to be presentable as \(UZ\) with \(H\)-unitary \(U\) and \(H\)-selfadjoint \(H\)-consistent \(Z\) (\(H\)- polar decomposition of the operator) are found. The paper contains sections of introduction, \(H\)-consistent operators and their properties, and \(H_1\)-\(H_2\)-consistent operator and its canonical form, the equation \(X^{[*]} X= A\), classification of eigensubspaces of nilpotent \(H\)-selfadjoint operator, classification of linear mappings from one space with indefinite scalar product to another, examples and introduction to \(H\)-polar decomposition.
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A63
zbMATH DE Number: 778241
zbMATH Keywords: unitary equivalence
zbMATH Keywords: indefinite scalar product
zbMATH Keywords: singular-value decomposition
zbMATH Keywords: consistent operator
zbMATH Keywords: polar decomposition
zbMATH Keywords: Hermitian matrix
zbMATH Keywords: Jordan form
zbMATH Keywords: canonical form
Wikidata QID: Q117378937
reviewed by: Yueh-er Kuo
MaRDI profile type: Publication
cites work: Q3871028
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description: scientific article; zbMATH DE number 778241