Unitary equivalence in an indefinite scalar product: An analogue of singular-value decomposition

AbstractIf H1 is an n × n and H2 an m × m invertible Hermitian matrix and X and Y are arbitrary complex m × n matrices, we call the last matrices equivalent if X = U2YU1 for some H1-unitary matrix U1 and some H2-unitary matrix U2. It is well known that if H1 and H2 are positive definite, then without loss of generality we can assume that they are identities, and X and Y are equivalent if and only if the (diagonalizable) matrices X∗X and Y∗Y have the same spectrum. In the present paper we show that, in general, the Jordan form of X[∗]X, where X[∗] is the H1-H2-adjoint of X,X[∗] = H−11X∗H2, 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-self-adjoint H-consistent Z (“H-polar decomposition of the operator”) are found