Nonnegative matrix

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 the interior of the set of all non-negative matrices. While such matrices are commonly found, the term "positive matrix" 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.