Nonnegative matrix

A nonnegative matrix is a matrix in which all the elements are equal to or greater than zero

$\mathbf {X} \geq 0,\qquad \forall _{ij}\,x_{ij}\geq 0.$

A positive matrix is a matrix in which all the elements are greater than zero. The set of positive matrices is a subset of all non-negative matrices.

A non-negative matrix can represent a transition matrix for a Markov chain.

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

A positive matrix is not the same as a positive-definite matrix. A matrix that is both non-negative and positive semidefinite is called a doubly non-negative matrix.

Eigenvalue and eigenvectors of squared positive matrices are described by the Perron–Frobenius theorem.

Inversion

An inverse of a non-singular so-called M-matrix 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, and so on.

Bibliography

• Abraham Berman, Robert J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, 1994, ISBN 0-89871-321-8.