{{Short description|Matrix with no negative elements}} {{hatnote|Not to be confused with Totally positive matrix and Positive-definite matrix.}}
In mathematics, a '''nonnegative matrix''', written : <math>\mathbf{X} \geq 0,</math> is a matrix in which all the elements are equal to or greater than zero, that is, : <math>x_{ij} \geq 0\qquad \forall {i,j}.</math> 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 {{Clarify|reason=relation to subject of nonnegative matrix not made clear; what is an M-matrix?|date=March 2015}} 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 {{math|''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 == {{refbegin}} * {{cite book |first1=Abraham |last1=Berman |first2=Robert J. |last2=Plemmons |author2-link=Robert J. Plemmons |title=Nonnegative Matrices in the Mathematical Sciences |publisher=SIAM |date=1994 |isbn=0-89871-321-8 |doi=10.1137/1.9781611971262}} *{{harvnb|Berman|Plemmons|1994|loc=2. Nonnegative Matrices pp. 26–62. {{doi|10.1137/1.9781611971262.ch2}}}} *{{cite book |first1=R.A. |last1=Horn |first2=C.R. |last2=Johnson |chapter=8. Positive and nonnegative matrices |title=Matrix Analysis |publisher=Cambridge University Press |edition=2nd |date=2013 |isbn=978-1-139-78203-6 |oclc=817562427 }} * {{cite book| last = Krasnosel'skii | first = M. A. | authorlink = Mark Krasnosel'skii | title=Positive Solutions of Operator Equations | publisher=P. Noordhoff | location= Groningen | year=1964 |oclc=609079647}} *{{cite book| last1 = Krasnosel'skii | first1 = M. A. | authorlink1=Mark Krasnosel'skii | last2 = Lifshits | first2 = Je.A. | last3 = Sobolev | first3 = A.V. | title = Positive Linear Systems: The method of positive operators | series = Sigma Series in Applied Mathematics | volume=5 | publisher = Helderman Verlag | isbn=3-88538-405-1 |oclc=1409010096 | year=1990}} * {{cite book |first=Henryk |last=Minc |title=Nonnegative matrices |publisher=Wiley |date=1988 |isbn=0-471-83966-3 |oclc=1150971811}} * {{cite book |author-link=Eugene Seneta |first=E. |last=Seneta |title=Non-negative matrices and Markov chains |publisher=Springer |series=Springer Series in Statistics |edition=2nd |date=1981 |isbn=978-0-387-29765-1 |oclc=209916821 |doi=10.1007/0-387-32792-4}} * {{cite book |author-link=Richard S. Varga |first=R.S. |last=Varga |chapter=Nonnegative Matrices |chapter-url=https://link.springer.com/chapter/10.1007/978-3-642-05156-2_2 |doi=10.1007/978-3-642-05156-2_2 |title=Matrix Iterative Analysis |publisher=Springer |series=Springer Series in Computational Mathematics |volume=27 |date=2009 |isbn=978-3-642-05156-2 |pages=31–62 }} * 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). {{refend}}
{{Matrix classes}}
Category:Matrices (mathematics)