Theoretical and computational properties of semi-infinite quasi-To eplitz M-matrices

A quasi-To eplitz 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) is an element of C, i, j >= 1 and E is a compact correction. Quasi-To eplitz 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-To eplitz matrix is an M-matrix. We show that under a mild and easy-to-check condition, an invertible quasi-To eplitz M-matrix has a unique square root that is an M-matrix possessing quasi-To eplitz structure. The quasi-To eplitz structure of the square root of M-matrices provides inspirations for proving spectral properties of the quadratic matrix polynomial L(lambda) = lambda(2)A(1)+ lambda A(0)+A(-1) having quasi-To eplitz coefficients where A(1), A(-1) >= 0 and -A(0) in addition is an M-matrix. Some issues concerning the computation of square root of quasi-To eplitz M-matrices are discussed and numerical experiments are performed. (C) 2022 Elsevier Inc. All rights reserved.