Gohberg-Semencul Type Formulas via Embedding of

We present a new way of deriving Gohberg-Se- mencul type inversion formulas for Hermitian Toeplitz and quasi-Toeplitz matrices. Our approach is based on certain C-lossless embedding of Lyapunov equations. It has been shown that if a nonsingular matrix R has Toeplitz displacement inertia { p, q} then R-' does not have the same Toeplitz displacement inertia. However, a para-Hermitian conjugate R-' (which is defined in Section I) of R-' will have this property. We have also shown that the Gohberg-Semencul type inversion formulas can be formed directly in terms of certain parameters of the embedding.