M-matrix

In mathematics, especially linear algebra, an M-matrix is a Z-matrix with eigenvalues whose real parts are nonnegative. The set of non-singular M-matrices are a subset of the class of P-matrices, and also of the class of inverse-positive matrices (i.e. matrices with inverses belonging to the class of positive matrices). The name M-matrix was seemingly originally chosen by Alexander Ostrowski in reference to Hermann Minkowski, who proved that if a Z-matrix has all of its row sums positive, then the determinant of that matrix is positive. An M-matrix is commonly defined as follows: Definition: Let A be a n × n real Z-matrix. That is, A = (aij) where aij ≤ 0 for all i ≠ j, 1 ≤ i,j ≤ n. Then matrix A is also an M-matrix if it can be expressed in the form A = sI − B, where B = (bij) with bij ≥ 0, for all 1 ≤ i,j ≤ n, where s is at least as large as the maximum of the moduli of the eigenvalues of B, and I is an identity matrix. For the non-singularity of A, according to the Perron–Frobenius theorem, it must be the case that s > ρ(B). Also, for a non-singular M-matrix, the diagonal elements aii of A must be positive. Here we will further characterize only the class of non-singular M-matrices. Many statements that are equivalent to this definition of non-singular M-matrices are known, and any one of these statements can serve as a starting definition of a non-singular M-matrix. For example, Plemmons lists 40 such equivalences. These characterizations has been categorized by Plemmons in terms of their relations to the properties of: (1) positivity of principal minors, (2) inverse-positivity and splittings, (3) stability, and (4) semipositivity and diagonal dominance. It makes sense to categorize the properties in this way because the statements within a particular group are related to each other even when matrix A is an arbitrary matrix, and not necessarily a Z-matrix. Here we mention a few characterizations from each category. Below, ≥ denotes the element-wise order (not the usual positive semidefinite order on matrices).