Gohberg-Semencul type formulas via embedding of Lyapunov equations [signal processing]

The authors present a new way of deriving Gohberg-Semencul-type inversion formulas for Hermitian Toeplitz and quasi-Toeplitz matrices. The approach is based on a certain Σ-lossless embedding of Lyapunov equations. It has been shown that if a nonsingular matrix R has Toeplitz displacement inertia {p, q}, then R^-1 does not have the same Toeplitz displacement inertia. However, a para-Hermitian conjugate of R^-1 will have this property. It is also shown that the Gohberg-Semencul-type inversion formulas can be formed directly in terms of certain parameters of the embedding