nonnegative matrix
{{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
:
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.
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
Bibliography
{{refbegin}}
- {{cite book |first=Abraham |last=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 |first=R.A. |last=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}}