Nonnegative matrix

In mathematics, a nonnegative matrix, written In mathematics, a nonnegative matrix, written is a matrix in which all the elements are equal to or greater than zero, that is, A positive matrix is a matrix in which all the elements are strictly greater than zero. The set of positive matrices is a subset of all non-negative matrices. While such matrices are commonly found, the term is only occasionally used due to the possible confusion with positive-definite matrices, which are different. A rectangular non-negative matrix can be approximated by a decomposition with two other non-negative matrices 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. Eigenvalues and eigenvectors of square positive matrices are described by the Perron–Frobenius theorem. 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. {displaystyle n>1.} 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.