Inversion formulas for Toeplitz-plus-Hankel matrices

Abstract

The main aim of the present paper is to establish inversion formulas of Gohberg-Semencul type for Toeplitz-plus-Hankel matrices. In particular, it is shown how the inverse of such a structured matrix An of order n is computed by means of their first two and last two columns or rows under the additional assumption that a certain 2×2 matrix is nonsingular. Moreover, a formula for the inverse of the Toeplitz-plus-Hankel matrix An−2 of order n−2 in the center of An is established and sufficient invertibility criteria for both An and An−2 are obtained. Hereby a main tool is to use known inversion formulas involving not only rows or columns of the inverse but also solutions of equations the right hand sides of which depend on entries of the Toeplitz-plus-Hankel matrix itself.