Nonnegative matrix: Difference between revisions

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{{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 (mathematics)|matrix]] in which all the elements are equal to or greater than zero, that is,

: <math>\mathbfx_{Xij} \geq 0, \qquad \forall {i,j\, x_{ij} \geq 0.</math>

A '''positive matrix''' is a matrix in which all the elements are strictly greater than zero. The set of positive matrices is athe subsetinterior 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 matrix|positive-definite matrices]], which are different. A matrix which is both non-negative and is positive semidefinite is called a '''doubly non-negative matrix'''.

Any [[transition matrix]] for a [[Markov chain]] is a 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]].

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]].

==Properties==

*The [[Trace (linear algebra)|trace]] and every row and column sum/product of a nonnegative matrix is nonnegative.

== Inversion ==

The inverse of any [[Invertible matrix|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}}.</math>

== Specializations ==

Line 23 ⟶ 24:

== See also ==

* [[Metzler matrix]]

[[Metzler matrix]]

== Bibliography ==

{{refbegin}}

# Abraham Berman, Robert J. Plemmons, ''Nonnegative Matrices in the Mathematical Sciences'', 1994, SIAM. ISBN 0-89871-321-8.

#A.* {{cite book |first1=Abraham |last1=Berman and|first2=Robert RJ. |last2=Plemmons |author2-link=Robert J. Plemmons, ''|title=Nonnegative Matrices in the Mathematical Sciences'', Academic|publisher=SIAM Press, 1979 (chapter 2), ISBN|date=1994 |isbn=0-1289871-092250321-98 |doi=10.1137/1.9781611971262}}

*{{harvnb|Berman|Plemmons|1994|loc=2. Nonnegative Matrices pp. 26–62. {{doi|10.1137/1.9781611971262.ch2}}}}

#R.A. Horn and C.R. Johnson, ''Matrix Analysis'', Cambridge University Press, 1990 (chapter 8).

*{{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 Ltd

| ___location= [[Groningen (city)|Groningen]]

| year=1964 | pagesoclc=381 pp.609079647}}

#*{{cite book| last1 = Krasnosel'skii

| first1 = M. A.

| authorlink1=Mark Krasnosel'skii

Line 45 ⟶ 46:

| first3 = A.V.

| title = Positive Linear Systems: The method of positive operators

| series = Sigma Series in Applied Mathematics | volume=5 |pages=354 pp.

| publisher = Helderman Verlag

| isbn=3-88538-405-1 |oclc=1409010096

| ___location= [[Berlin]]

| year=1990}}

#* {{cite book |first=Henryk |last=Minc, ''|title=Nonnegative matrices'', John |publisher=Wiley&Sons, New York, |date=1988, ISBN |isbn=0-471-83966-3 |oclc=1150971811}}

#* {{cite book |author-link=Eugene Seneta, |first=E. ''|last=Seneta |title=Non-negative matrices and Markov chains''. 2nd rev. ed., 1981, XVI, 288 p., Softcover|publisher=Springer |series=Springer Series in Statistics. (Originally|edition=2nd published by Allen & Unwin Ltd., London, 1973) ISBN|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 }}

# [[Richard S. Varga]] 2002 ''Matrix Iterative Analysis'', Second ed. (of 1962 Prentice Hall edition), Springer-Verlag.

* 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}}

[[Category:Matrices]]

{{Matrix classes}}

{{Linear-algebra-stub}}

[[Category:Matrices (mathematics)]]

[[sl:Nenegativna matrika]]

[[sv:Icke-negativ matris]]