Theoretical and computational properties of quasi-Toeplitz M-matrices

by Dr Jie Meng (Ocean University of China)

Monday 15 Jul 2024, 15:00 → 16:00 Europe/Rome

Sala Riunioni (Dipartimento di Matematica)

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