M-matrix

M-matrix

An M-matrix is a square matrix A that satisfies the condition that there exists a positive vector v such that Av ≥ 0, or alternatively, that the off-diagonal entries are non-positive and the real eigenvalues (as well as real parts of all eigenvalues) are nonnegative. Additionally, an M-matrix is known as a Stieltjes matrix and is particularly stable in numerical methods such as incomplete Cholesky factorization.From: Group Inverses of M-Matrices and Their Applications [2019], Tolerances, robustness and parametrization of matrix properties related to optimization problems [2018], Delaunay Mesh Generation [2019]

Nonnegative Stabilization of Interval Discrete Systems

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Published in S.P. Bhattacharyya, L.H. Keel, of UNCERTAIN DYNAMIC SYSTEMS, 2020

B. Shafai, C. V. Hollot

As we have pointed out before the class of M-matrices, although they are not nonnegative, are closely related to nonnegative matrices with remarkable properties. The main diagonal entries of an M-matrix are nonnegative and all its other entries are nonpositive. Furthermore the principal minors and the inverse of an M-matrix are nonnegative. Since all the eigenvalues of an M-matrix have nonnegative real parts, the nagative of an M-matrix becomes a stable Metzler matrix which establishes a direct link to the stability of continuous time systems.

Tolerances, robustness and parametrization of matrix properties related to optimization problems

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Published in Optimization, 2018

Milan Hladík

Parametrization – exact bounds. By definition, is an M-matrix if the off-diagonal entries are non-positive and the real eigenvalues (as well as real parts of all eigenvalues) are nonnegative. The former is handled easily, and for the latter we simply control that stays positive. Due to continuity of eigenvalues, this implies that all real eigenvalues (and real parts of all eigenvalues) stay positive. Therefore, we handle this problem by the techniques from Section 2.

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Nonnegative Stabilization of Interval Discrete Systems

Tolerances, robustness and parametrization of matrix properties related to optimization problems