# Nonnegative matrix

__: Matrix whose elements are all ≥0__

**Short description**In mathematics, a **nonnegative matrix**, written

- [math]\displaystyle{ \mathbf{X} \geq 0, }[/math]

is a matrix in which all the elements are equal to or greater than zero, that is,

- [math]\displaystyle{ x_{ij} \geq 0\qquad \forall {i,j}. }[/math]

A **positive matrix** is a matrix in which all the elements are strictly greater than zero. The set of positive matrices is a subset of all non-negative matrices. While such matrices are commonly found, the term is only occasionally used due to the possible confusion with positive-definite matrices, which are different. A matrix which is both non-negative and is positive semidefinite is called a **doubly non-negative matrix**.

A rectangular non-negative matrix can be approximated by a decomposition with two other non-negative matrices via non-negative matrix factorization.

Eigenvalues and eigenvectors of square positive matrices are described by the Perron–Frobenius theorem.

## Properties

- The trace and every row and column sum/product of a nonnegative matrix is nonnegative.

## Inversion

The inverse of any non-singular M-matrix^{[clarification needed]} is a non-negative matrix. If the non-singular M-matrix is also symmetric then it is called a Stieltjes matrix.

The inverse of a non-negative matrix is usually not non-negative. The exception is the non-negative monomial matrices: a non-negative matrix has non-negative inverse if and only if it is a (non-negative) monomial matrix. Note that thus the inverse of a positive matrix is not positive or even non-negative, as positive matrices are not monomial, for dimension *n* > 1.

## Specializations

There are a number of groups of matrices that form specializations of non-negative matrices, e.g. stochastic matrix; doubly stochastic matrix; symmetric non-negative matrix.

## See also

## Bibliography

- Abraham Berman, Robert J. Plemmons,
*Nonnegative Matrices in the Mathematical Sciences*, 1994, SIAM. ISBN 0-89871-321-8. - A. Berman and R. J. Plemmons,
*Nonnegative Matrices in the Mathematical Sciences*, Academic Press, 1979 (chapter 2), ISBN 0-12-092250-9 - R.A. Horn and C.R. Johnson,
*Matrix Analysis*, Cambridge University Press, 1990 (chapter 8). - Krasnosel'skii, M. A. (1964).
*Positive Solutions of Operator Equations*. Groningen: P.Noordhoff Ltd. pp. 381 pp. - Krasnosel'skii, M. A.; Lifshits, Je.A.; Sobolev, A.V. (1990).
*Positive Linear Systems: The method of positive operators*. Sigma Series in Applied Mathematics.**5**. Berlin: Helderman Verlag. pp. 354 pp. - Henryk Minc,
*Nonnegative matrices*, John Wiley&Sons, New York, 1988, ISBN 0-471-83966-3 - Seneta, E.
*Non-negative matrices and Markov chains*. 2nd rev. ed., 1981, XVI, 288 p., Softcover Springer Series in Statistics. (Originally published by Allen & Unwin Ltd., London, 1973) ISBN 978-0-387-29765-1 - Richard S. Varga 2002
*Matrix Iterative Analysis*, Second ed. (of 1962 Prentice Hall edition), Springer-Verlag.

Original source: https://en.wikipedia.org/wiki/Nonnegative matrix.
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