**Date and time.**

Monday, Jul 15, 2024 15:00-16:00

Aula Riunioni (Department of Mathematics) Seminars, Seminar on Numerical Analysis

**Abstract.**

A quasi-Toeplitz matrix is a semi-infinite matrix of the form $A=T(a)+E$, where $T(a)$ is a Toeplitz matrix with entries $(T(a))_{i,j}=a_{j-i}$, for $a_{j-i}\in\mathbb C$, $i,j\ge 1$ and $E$ is a compact correction. Quasi-Toeplitz $M$-matrices are encountered in the study of quadratic matrix equations arising in the analysis of a 2-dimensional Quasi-Birth-Death (QBD) stochastic process. We investigate the properties of such matrices and provide conditions under which a quasi-Toeplitz matrix is an $M$-matrix. We show that under a mild and easy-to-check condition, an invertible quasi-Toeplitz $M$-matrix has a unique square root that is an $M$-matrix possessing quasi-Toeplitz structure. Some issues concerning the computation of the square root of quasi-Toeplitz $M$-matrices are discussed and numerical experiments are performed. Indico event.