# Nonnegative matrix

Short description: Matrix whose elements are all ≥0

In mathematics, a nonnegative matrix, written

$\displaystyle{ \mathbf{X} \geq 0, }$

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

$\displaystyle{ x_{ij} \geq 0\qquad \forall {i,j}. }$

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.