Nonnegative matrix

Not to be confused with Totally positive matrix and Positive-definite matrix.

In mathematics, a nonnegative matrix, written

$𝐗\ge 0,$

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

$x_{ij}\ge 0\mathrm{\forall }i,j.$

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.

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 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

• Metzler matrix

Bibliography

• Berman & Plemmons 1994, 2. Nonnegative Matrices pp. 26–62.
• Andrzej Cichocki; Rafel Zdunek; Anh Huy Phan; Shun-ichi Amari: Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation, John Wiley & Sons,ISBN 978-0-470-74666-0 (2009).