# M-matrix

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Matrix in mathematics

In [mathematics](/source/Mathematics), especially [linear algebra](/source/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](/source/Z-matrix_(mathematics))) and whose [eigenvalues](/source/Eigenvalue) have nonnegative [real parts](/source/Complex_number#Notation). The set of non-singular *M*-matrices are a subset of the class of [*P*-matrices](/source/P-matrix), and also of the class of [inverse-positive matrices](/source/Inverse-positive_matrix) (i.e. matrices with inverses belonging to the class of [positive matrices](/source/Nonnegative_matrix)).[1] The name *M*-matrix was seemingly originally chosen by [Alexander Ostrowski](/source/Alexander_Ostrowski) in reference to [Hermann Minkowski](/source/Hermann_Minkowski), who proved that if a Z-matrix has all of its row sums positive, then the determinant of that matrix is positive.[2]

## Characterizations

An M-matrix is commonly defined as follows:

**Definition:** Let *A* be a *n* × *n* real [Z-matrix](/source/Z-matrix_(mathematics)). 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](/source/Non-singularity) of *A*, according to the [Perron–Frobenius theorem](/source/Perron%E2%80%93Frobenius_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.[3] For example, Plemmons lists 40 such equivalences.[4] 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.

## Properties

Below, ≥ denotes the element-wise order (not the usual [positive semidefinite](/source/Positive_semidefinite_matrix) order on matrices). That is, for any real matrices *A*, *B* of size *m* × *n*, we write *A* ≥ *B* (or *A* > *B*) if *a**ij* ≥ *b**ij* (or *a**ij* > *b**ij*) for all *i*, *j*.

Let *A* be a *n* × *n* real [Z-matrix](/source/Z-matrix_(mathematics)), then the following statements are equivalent to *A* being a [non-singular](/source/Algebraic_curve#Singularities) M-matrix:

*Positivity of principal minors*

- All the [principal minors](/source/Minor_(linear_algebra)) of *A* are positive. That is, the determinant of each submatrix of *A* obtained by deleting a set, possibly empty, of corresponding rows and columns of *A* is positive.

- *A* + *D* is non-singular for each nonnegative diagonal matrix *D*.

- Every real eigenvalue of *A* is positive.

- All the leading principal minors of *A* are positive.

- There exist lower and [upper triangular](/source/Triangular_matrix) matrices *L* and *U* respectively, with positive diagonals, such that *A* = *LU*.

*Inverse-positivity and splittings*

- *A* is *inverse-positive*. That is, *A*−1 exists and *A*−1 ≥ 0.

- *A* is *monotone*. That is, *Ax* ≥ 0 implies *x* ≥ 0.

- *A* has a *convergent regular splitting*. That is, *A* has a representation *A* = *M* − *N*, where *M*−1 ≥ 0, *N* ≥ 0 with *M*−1*N* *convergent*. That is, *ρ*(*M*−1*N*) < 1.

- There exist inverse-positive matrices *M*1 and *M*2 with *M*1 ≤ *A* ≤ *M*2.

- Every regular splitting of *A* is convergent.

*Stability*

- There exists a positive diagonal matrix *D* such that *AD* + *DAT* is positive definite.

- *A* is *positive stable*. That is, the real part of each eigenvalue of *A* is positive.

- There exists a symmetric [positive definite matrix](/source/Positive_definite_matrix) *W* such that *AW* + *WAT* is positive definite.

- *A* + *I* is non-singular, and *G* = (*A* + *I*)−1(*A* − *I*) is convergent.

- *A* + *I* is non-singular, and for *G* = (*A* + *I*)−1(*A* − *I*), there exists a positive definite symmetric matrix *W* such that *W* − *GTWG* is positive definite.

*Semipositivity and diagonal dominance*

- *A* is *semi-positive*. That is, there exists *x* > 0 with *Ax* > 0.

- There exists *x* ≥ 0 with *Ax* > 0.

- There exists a positive diagonal matrix *D* such that AD has all positive row sums.

- *A* has all positive diagonal elements, and there exists a positive diagonal matrix *D* such that AD is *strictly [diagonally dominant](/source/Diagonally_dominant)*.

- *A* has all positive diagonal elements, and there exists a positive diagonal matrix *D* such that *D*−1*AD* is strictly diagonally dominant.

## Applications

The primary contributions to M-matrix theory has mainly come from mathematicians and economists. M-matrices are used in mathematics to establish bounds on eigenvalues and on the establishment of convergence criteria for [iterative methods](/source/Iterative_methods) for the solution of large [sparse](/source/Sparse_matrix) [systems of linear equations](/source/Systems_of_linear_equations). M-matrices arise naturally in some discretizations of [differential operators](/source/Differential_operators), such as the [Laplacian](/source/Laplacian), and as such are well-studied in scientific computing. M-matrices also occur in the study of solutions to [linear complementarity problem](/source/Linear_complementarity_problem). Linear complementarity problems arise in [linear](/source/Linear_programming) and [quadratic programming](/source/Quadratic_programming), [computational mechanics](/source/Computational_mechanics), and in the problem of finding [equilibrium point](/source/Equilibrium_point) of a [bimatrix game](/source/Bimatrix_game). Lastly, M-matrices occur in the study of finite [Markov chains](/source/Markov_chains) in the field of [probability theory](/source/Probability_theory) and [operations research](/source/Operations_research) like [queuing theory](/source/Queuing_theory). Meanwhile, economists have studied M-matrices in connection with gross substitutability, stability of a [general equilibrium](/source/General_equilibrium_theory) and [Leontief's input–output analysis](/source/Input%E2%80%93output_model) in economic systems. The condition of positivity of all principal minors is also known as the Hawkins–Simon condition in economic literature.[5] In engineering, M-matrices also occur in the problems of [Lyapunov stability](/source/Lyapunov_stability) and [feedback control](/source/Feedback_control) in [control theory](/source/Control_theory) and are related to [Hurwitz matrices](/source/Hurwitz-stable_matrix). In [computational biology](/source/Computational_biology), M-matrices occur in the study of [population dynamics](/source/Population_dynamics).

## See also

- A is a non-singular weakly [diagonally dominant](/source/Diagonally_dominant) M-matrix if and only if it is a [weakly chained diagonally dominant](/source/Weakly_chained_diagonally_dominant) [L-matrix](/source/L-matrix).

- If A is an M-matrix, then −A is a [Metzler matrix](/source/Metzler_matrix).

- A non-singular symmetric *M*-matrix is sometimes called a [Stieltjes matrix](/source/Stieltjes_matrix).

- [Translation matrix](/source/Translation_(geometry)#Matrix_representation)

- [Hurwitz-stable matrix](/source/Hurwitz-stable_matrix)

- [P-matrix](/source/P-matrix)

- [Perron–Frobenius theorem](/source/Perron%E2%80%93Frobenius_theorem)

- [Z-matrix](/source/Z-matrix_(mathematics))

- [H-matrix](/source/H-matrix_(iterative_method))

## References

1. **[^](#cite_ref-1)** Fujimoto, Takao & Ranade, Ravindra (2004), ["Two Characterizations of Inverse-Positive Matrices: The Hawkins-Simon Condition and the Le Chatelier-Braun Principle"](http://www.emis.de/journals/ELA/ela-articles/articles/vol11_pp59-65.pdf) (PDF), *Electronic Journal of Linear Algebra*, **11**: 59–65, [doi](/source/Doi_(identifier)):[10.13001/1081-3810.1122](https://doi.org/10.13001%2F1081-3810.1122).

1. **[^](#cite_ref-Berman_2-0)** Bermon, Abraham; [Plemmons, Robert J.](/source/Robert_J._Plemmons) (1994), *Nonnegative Matrices in the Mathematical Sciences*, Philadelphia: Society for Industrial and Applied Mathematics, p. 134,161 (Thm. 2.3 and Note 6.1 of chapter 6), [ISBN](/source/ISBN_(identifier)) [0-89871-321-8](https://en.wikipedia.org/wiki/Special:BookSources/0-89871-321-8).

1. **[^](#cite_ref-3)** Fiedler, M; Ptak, V. (1962), "On matrices with non-positive off-diagonal elements and positive principal minors", *Czechoslovak Mathematical Journal*, **12** (3): 382–400, [doi](/source/Doi_(identifier)):[10.21136/CMJ.1962.100526](https://doi.org/10.21136%2FCMJ.1962.100526), [hdl](/source/Hdl_(identifier)):[10338.dmlcz/100526](https://hdl.handle.net/10338.dmlcz%2F100526).

1. **[^](#cite_ref-4)** Plemmons, R.J. (1977), "M-Matrix Characterizations. I -- Nonsingular M-Matrices", *Linear Algebra and Its Applications*, **18** (2): 175–188, [doi](/source/Doi_(identifier)):[10.1016/0024-3795(77)90073-8](https://doi.org/10.1016%2F0024-3795%2877%2990073-8).

1. **[^](#cite_ref-5)** Nikaido, H. (1970). *Introduction to Sets and Mappings in Modern Economics*. New York: Elsevier. pp. 13–19. [ISBN](/source/ISBN_(identifier)) [0-444-10038-5](https://en.wikipedia.org/wiki/Special:BookSources/0-444-10038-5).

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