An extension problem for H-unitary matrices with applications to Hermitian Toeplitz matrices (Q1124657)

A Hermitian matrix H and a rectangular matrix \(U_ 0\) are given such that \(U^ H_ 0HU_ 0=A\), where A is the leading principal submatrix of H; the extension problem of \(U_ 0\) to an H-unitary matrix U is studied. The solvability conditions known for a nonsingular matrix H are derived for the case of singular H. Necessary conditions are presented for a more general problem described by means of Hermitian linear operators on linear spaces with inner products. The complete solution, that means a necessary and sufficient criterion, is derived for the matrix problem together with a description of all its solutions. These results are used to obtain some properties of factorizations of Hermitian Toeplitz matrices.