Theoretical and computational properties of semi-infinite quasi-Toeplitz M-matrices

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=aj−i, for aj−i∈C, i,j≥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 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. The quasi-Toeplitz structure of the square root of M-matrices provides inspirations for proving spectral properties of the quadratic matrix polynomial L(λ)=λ2A1+λA0+A−1 having quasi-Toeplitz coefficients where A1,A−1≥0 and −A0 in addition is an M-matrix. Some issues concerning the computation of square root of quasi-Toeplitz M-matrices are discussed and numerical experiments are performed.