The explicit inverse of nonsingular conjugate-Toeplitz and conjugate-Hankel matrices (Q513478)

Toeplitz matrices have important applications in many fields, e.\,g.\ communications, image processing and signal processing. The derivation of the inverse of a Toeplitz matrix is an important problem and numerous authors have devised algorithms to do so, using differing parts of the given Toeplitz matrix. Generalized inverses have also been studied. \textit{S. Barnett} and \textit{M. J. C. Gover} [Linear Multilinear Algebra 14, 45--65 (1983; Zbl 0536.15011)] defined a square matrix \(T\) to be conjugate-Toeplitz (CT) if \(t_{i+1,\,j+1}\) is the complex conjugate of \(t_{i,\,j}\) for all \(i\),\,\(j\), and proposed an algorithm for inverting strongly nonsingular CT matrices. They also defined a square matrix \(H\) to be conjugate-Hankel (CH) if \(t_{i+1,\,j}\) is the complex conjugate of \(t_{i,\,j+1}\) for all \(i\), \(j\). In this paper, the authors give two algorithms for inverting CT matrices, and similarly for CH matrices. They prove that these inverses are expressible as the sum of products of lower and upper triangular matrices. They also study the stability of these algorithms. The paper concludes with two numerical examples.