M-matrix
In mathematics, particularly in the field of linear algebra, an M-matrix is a matrix that satisfies certain properties. The term "M-matrix" generally refers to a real square matrix A that can be expressed in the form A = sI - B, where B is a nonnegative matrix (all entries are greater than or equal to zero), s is a real number greater than or equal to the spectral radius of B (the largest absolute value of its eigenvalues), and I is the identity matrix.
More formally, an M-matrix fulfills the following conditions:
- It is a real square matrix.
- It can be written as A = sI - B, where B is a nonnegative matrix.
- s ≥ ρ(B), where ρ(B) is the spectral radius of B.
M-matrices have several important properties and are used in a variety of applications, including:
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Stability Analysis: M-matrices are linked to the stability of dynamical systems and iterative processes. If a matrix related to a system is an M-matrix, it often indicates that the system is stable.
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Numerical Analysis: They appear in the analysis of iterative methods for solving linear systems, particularly in the context of finite difference and finite element methods.
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Mathematical Economics: M-matrices are used in models of economic equilibrium and input-output analysis.
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Markov Chains: M-matrices are related to the study of Markov chains, particularly in the context of transition rate matrices.
Various equivalent definitions and characterizations of M-matrices exist. One common characterization involves the concept of "regular splitting." A splitting A = M - N of a matrix A is called a regular splitting if M is nonsingular, M-1 is nonnegative, and N is nonnegative. A matrix A is an M-matrix if and only if it has a regular splitting A = M - N where M is a nonsingular M-matrix.
Furthermore, an equivalent characterization of an M-matrix A is that it is nonsingular and A-1 is nonnegative. This property makes them crucial in applications where nonnegativity is important.