Inversion formulas for Toeplitz-plus-Hankel matrices (Q6565850)

The main motivation for the questions considered in this paper comes from results by \textit{I. C. Gohberg} and \textit{A. A. Semencul} [Mat. Issled. 7, No. 2(24), 201--223 (1972; Zbl 0288.15004)], who derived an inversion formula for Toeplitz matrices \[T_n(\textbf{a}) = [a_{i-j}]_{i,j=0}^{n-1}, \qquad \textbf{a}=(a_j)_{j=-n+1}^{n-1}.\] \NThis formula expresses \(T_n(\textbf{a})^{-1}\) as a sum of products of triangular Toeplitz matrices involving only the entries of the first and last column (or row) of \(T_n(\textbf{a})^{-1}\). Such inversion formulas can also be established for Hankel matrices \[H_n(\textbf{b}) = [b_{i+j-n+1}]_{i,j=0}^{n-1}, \qquad \textbf{b}=(b_j)_{j=-n+1}^{n-1}.\]\N\NThe main goal of the present paper is to establish inversion formulas of Gohberg-Semencul type for matrices which are the sum of a Toeplitz and a Hankel matrix (such matrices are called \textit{Toeplitz-plus-Hankel matrices} or \(T+H\) matrices), i.e., \[ TH_n(\textbf{a}, \textbf{b}) = T_n(\textbf{a}) + H_n(\textbf{b}).\]\N\NThe main results are presented in Section 3 and Section 4. In the first of these two sections, they authors obtain invertibility criteria for the \(n \times n\) \(T+H\) matrix \(TH_n(\textbf{a}, \textbf{b})\) and establish recursion formulas for all columns (or rows) of \(TH_n(\textbf{a}, \textbf{b})^{-1}\). These results are not new, but the direct proof is original. The second of these two sections begins with a reminder of a result found by \textit{G. Heinig} and the second author [Linear Algebra Appl. 106, 39--52 (1988; Zbl 0646.15015)], stating that the inverse of an invertible \(T+H\) matrix is an invertible \(T+H\)-Bezoutian, and {viceversa}. A matrix \(B= [b_{ij}]_{i,j=0}^{n-1}\) is called a \textit{Toeplitz-plus-Hankel-Bezoutian} (henceforth \(T+H\)-Bezoutian) if there exist eight vectors \(\textbf{u}_i, \textbf{v}_i \in \mathbb{F}^{n+2} (i = 1, 2, 3, 4)\) such that, in polynomial language, \[ B(t,s) = \sum\limits_{i,j=0}^{n-1}b_{ij}t^i s^j = \frac{\sum\limits_{i=1}^4 \textbf{u}_i(t) \textbf{v}_i(s)}{(t-s)(1-ts)}.\] Following this, the authors derive invertibility criteria and establish inversion formulas for \(TH_n(\textbf{a}, \textbf{b})\), where only the entries of four columns (or/and four rows) of \(TH_n(\textbf{a}, \textbf{b})^{-1}\) are involved. This is however shown to come at the expense that a certain \(2 \times 2\) matrix has to be nonsingular. Lastly, the authors derive inversion formulas of Gohberg-Semencul type for the \((n - 2) \times (n - 2)\) \(T+H\) matrix obtained from \(TH_n(\textbf{a}, \textbf{b})\) by deleting the first and the last column and the first and the last row\N\NIn Section 5 we have a simple example of size \(n = 5\) illustrating the recursion formula for the columns of \(TH_n(\textbf{a}, \textbf{b})^{-1}\) as well as some results stated in Section 4.