M-matrix

In mathematics, especially linear algebra, an "M"-matrix is a Z-matrix with eigenvalues whose real parts are positive. "M"-matrices are a subset of the class of "P"-matrices, and also of the class of inverse-positive matrices [Takao Fujimoto and Ravindra Ranade, "Two Characterizations of Inverse-Positive Matrices: The Hawkins-Simon Condition and the Le Chatelier-Braun Principle", "Electronic Journal of Linear Algebra" 11:59-65 (2004)] (i.e. matrices with inverses belonging to the class of positive matrices). A common characterization of "M"-matrices are square matrices with non-positive off-diagonal entries, positive diagonal entries, non-negative row sums, and at least one positive row sum.

The name "M"-matrix was seemingly originally chosen by Alexander Ostrowski in reference to Hermann Minkowski [Abraham Bermon and Robert J. Plemmons, "Nonnegative Matrices in the Mathematical Sciences", p. 161 (Note 6.1 of chapter 6)] .

A symmetric "M"-matrix is sometimes called a Stieltjes matrix.

"M"-matrices arise naturally in some discretizations of differential operators, particularly those with a minimum/maximum principle, such as the Laplacian, and as such are well-studied in scientific computing.

The "LU" factors of an "M"-matrix are guaranteed to exist and can be stably computed without need for numerical pivoting, also have positive diagonal entries and non-positive off-diagonal entries. Furthermore, this holds even for incomplete "LU" factorization, where entries in the factors are discarded during factorization, providing useful preconditioners for iterative solution.

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