A constructive proof of the Gohberg-Semencul formula (Q1123946)

\textit{I. C. Gohberg} and \textit{A. A. Semencul} [Mat. Issled. 7 Nr. 2(24), 201-223 (1972; Zbl 0288.15004)] gave some elegant formulas for the inverse of a Toeplitz matrix as a difference of products of lower and upper triangular Toeplitz matrices. There are several algebraic and analytic proofs of these formulas. Here the authors give a ``constructive'' proof for two of the Gohberg-Semencul formulas, under the assumption that the matrices are strongly nonsingular, i.e. all leading minors are nonzero. The procedure involves construction of a J- unitary matrix, postmultiplication by which zeros out the (1,3)-block of a matrix array constructed from the data of the problem. This J-unitary matrix in turn can be constructed as a product of elementary hyperbolic Householder reflections.