Unitary equivalence in an indefinite scalar product: An analogue of singular-value decomposition
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.
scientific article; zbMATH DE number 778241
scientific article