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Matrix whose elements are all ≥0 From Wikipedia, the free encyclopedia

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 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.

- The trace and every row and column sum/product of a nonnegative matrix is nonnegative.

The inverse of any non-singular M-matrix ^{[clarification needed]} 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.

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.

- Berman, Abraham; Plemmons, Robert J. (1994).
*Nonnegative Matrices in the Mathematical Sciences*. SIAM. doi:10.1137/1.9781611971262. ISBN 0-89871-321-8. - Berman & Plemmons 1994, 2. Nonnegative Matrices pp. 26–62. doi:10.1137/1.9781611971262.ch2
- Horn, R.A.; Johnson, C.R. (2013). "8. Positive and nonnegative matrices".
*Matrix Analysis*(2nd ed.). Cambridge University Press. ISBN 978-1-139-78203-6. OCLC 817562427. - Krasnosel'skii, M. A. (1964).
*Positive Solutions of Operator Equations*. Groningen: P. Noordhoff. OCLC 609079647. - Krasnosel'skii, M. A.; Lifshits, Je.A.; Sobolev, A.V. (1990).
*Positive Linear Systems: The method of positive operators*. Sigma Series in Applied Mathematics. Vol. 5. Helderman Verlag. ISBN 3-88538-405-1. OCLC 1409010096. - Minc, Henryk (1988).
*Nonnegative matrices*. Wiley. ISBN 0-471-83966-3. OCLC 1150971811. - Seneta, E. (1981).
*Non-negative matrices and Markov chains*. Springer Series in Statistics (2nd ed.). Springer. doi:10.1007/0-387-32792-4. ISBN 978-0-387-29765-1. OCLC 209916821. - Varga, R.S. (2009). "Nonnegative Matrices".
*Matrix Iterative Analysis*. Springer Series in Computational Mathematics. Vol. 27. Springer. pp. 31–62. doi:10.1007/978-3-642-05156-2_2. ISBN 978-3-642-05156-2.

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