M-matrix

In mathematics, especially linear algebra, an M-matrix is a matrix whose off-diagonal entries are less than or equal to zero (i.e., it is a Z-matrix) and whose eigenvalues have nonnegative real parts. 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).

Definition

Let A be an n × n real Z-matrix, meaning A = (a_ij) where a_ij ≤ 0 for all i ≠ j. Then A is an M-matrix if it can be expressed in the form A = sI − B, where B = (b_ij) with b_ij ≥ 0 for all i,j, s is at least as large as the maximum of the moduli of the eigenvalues of B, and I is the identity matrix.

For a non-singular M-matrix, it must be the case that s > ρ(B) (where ρ is the spectral radius), and the diagonal elements a_ii must be positive.

History

The name "M-matrix" was originally chosen by mathematician 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.

Key Characterizations

There are many equivalent characterizations of non-singular M-matrices (Plemmons lists 40 such equivalences). These include:

Positivity of principal minors:

• All principal minors of A are positive.
• Every real eigenvalue of A is positive.
• A + D is non-singular for each nonnegative diagonal matrix D.

Inverse-positivity and splittings:

• A is inverse-positive: A⁻¹ exists and A⁻¹ ≥ 0 (element-wise).
• A is monotone: Ax ≥ 0 implies x ≥ 0.
• A has a convergent regular splitting.

Stability:

• A is positive stable (the real part of each eigenvalue is positive).
• There exists a positive diagonal matrix D such that AD + DA^T is positive definite.

Semipositivity and diagonal dominance:

• There exists x > 0 with Ax > 0.
• A has all positive diagonal elements, and there exists a positive diagonal matrix D such that AD is strictly diagonally dominant.

Properties

• Every principal submatrix of a non-singular M-matrix is also a non-singular M-matrix.
• A non-singular symmetric M-matrix is sometimes called a Stieltjes matrix.
• If A is an M-matrix, then −A is a Metzler matrix.

Applications

M-matrices arise in numerous fields:

• Scientific computing: In discretizations of differential operators (such as the Laplacian) and in establishing convergence criteria for iterative methods for solving large sparse systems of linear equations.
• Economics: In connection with gross substitutability, stability of general equilibrium, and Leontief's input-output analysis. The condition of positivity of all principal minors is known as the Hawkins–Simon condition.
• Probability and operations research: In the study of finite Markov chains and queuing theory.
• Control theory: In Lyapunov stability analysis and feedback control, where M-matrices are related to Hurwitz matrices.
• Computational biology: In the study of population dynamics.
• Optimization: In the study of solutions to linear complementarity problems, which arise in linear and quadratic programming and computational mechanics.