UNITARY EQUIVALENCE IN AN INDEFINITE SCALAR PRODUCT - AN ANALOG OF SINGULAR-VALUE DECOMPOSITION

If H-1 is an n x n and H-2 an m x m invertible Hermitian matrix and X and Y are arbitrary complex m x n matrices, we call the last matrices equivalent if X = U2YU1 for some H-1-unitary matrix U-1 and some H-2-unitary matrix U-2. It is well known that if H-1 and H-2 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 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-self-adjoint H-consistent Z (''H-polar decomposition of the operator'') are found.