Abstract
One of the most important properties of M-matrices is element-wise non-negative of its inverse. In this paper, we consider element-wise perturbations of tridiagonal M-matrices and obtain bounds on the perturbations so that the non-negative inverse persists. The largest interval is given by which the diagonal entries of the inverse of tridiagonal M-matrices can be perturbed without losing the property of total nonnegativity. A numerical example is given to illustrate our findings.
Highlights
In many mathematical problems, Z -matrices and M -matrices play an important role
=β {β1, βk } ∈ Qk,n, we denote by A α β the k × k submatrix of A contained in the rows indexed by α1,αk and columns indexed by β1, βk
We start with some basic facts on tridiagonal M-matrices
Summary
Z -matrices and M -matrices play an important role. It is often useful to know the properties of their inverses, especially when the Z -matrices and the M-matrices have a special combinatorial structure, for more details we refer the reader [1]. One of the most important properties of some kinds of M-matrices is the nonegativity of their inverses, which plays central role in many of mathematical problems. (2015) Inverse Nonnegativity of Tridiagonal M-Matrices under Diagonal Element-Wise Perturbation. A noticeable amount of attention has turned to the inverse of tridiagonal M-matrices (those matrices which happen to be inverses of special form of M-matrices with property aij = 0 whenever i − j > 1 ) and M is generalized strictly diagonally dominant. A matrix is said to be generalized (strictly) diagonally dominant n if mii > ∑ mij. M is an M-matrix if and only if mij ≤ 0 , i ≠ j and mii > 0 , and M is generalized strictly diagonally dominant. We consider the inverse of perturbed M-matrix.
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