“Nonnegative matrix”的意思、由来-开放百科全书

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Nonnegative matrix

释义

Inversion
Specializations
See also
Bibliography

{{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 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.

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

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.

Bibliography

Abraham Berman, Robert J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, 1994, SIAM. {{isbn|0-89871-321-8}}.
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, 1979 (chapter 2), {{isbn|0-12-092250-9}}
R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 1990 (chapter 8).
{{cite book| last = Krasnosel'skii

| first = M. A.
| authorlink = Mark Krasnosel'skii
| title=Positive Solutions of Operator Equations
| publisher=P.Noordhoff Ltd
| location= Groningen
| year=1964| pages=381 pp.}}

{{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 |pages=354 pp.
| publisher = Helderman Verlag
| location= Berlin
| year=1990}}

Henryk Minc, Nonnegative matrices, John Wiley&Sons, New York, 1988, {{isbn|0-471-83966-3}}
Seneta, E. Non-negative matrices and Markov chains. 2nd rev. ed., 1981, XVI, 288 p., Softcover Springer Series in Statistics. (Originally published by Allen & Unwin Ltd., London, 1973) {{isbn|978-0-387-29765-1}}
Richard S. Varga 2002 Matrix Iterative Analysis, Second ed. (of 1962 Prentice Hall edition), Springer-Verlag.

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Inversion

Specializations

See also

Bibliography