On group inverse of singular Toeplitz matrices
The paper deals with representations of the group inverses of real singular Toeplitz matrices. The authors study the group inverse of the following \((n+1) \times (n+1)\) real singular Toeplitz matrix \[ M= \left[ \begin{matrix} A & c \\ b^T & d \end{matrix} \right], \] where \(A\) ia a real \(n \times n\) singular Toeplitz matrix, \(b\) and \(c\) are real \(1 \times n\) vectors and \(b^T\) denotes the transpose of \(b\). The main result of this paper is to show that if \(A\) and \(M\), in the previous expression, are singular matrices and have index one, then the group inverse of \(M\) can be represented as the sum of products of lower and upper triangular Toeplitz matrices. Such a matrix representation generalizes the matrix representation of the inverse known as ``Gohberg-Semencul formula''.
scientific article; zbMATH DE number 2160183