Nonnegative matrix

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A nonnegative matrix is a matrix where all the elements are equal to or above zero

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

A positive matrix is defined similarly. The set of positive matrices is a subset of all nonnegative matrices.

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

A rectangular nonnegative matrix can be approximated by a decomposition with two other nonnegative 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.

[edit] 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. An exception is the non-negative monomial matrices.

[edit] 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.