In mathematics, a nonnegative matrix, written
X ≥ 0 , {\displaystyle \mathbf {X} \geq 0,}is a matrix in which all the elements are equal to or greater than zero, that is,
x i j ≥ 0 ∀ i , j . {\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.
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
Bibliography
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- 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).