In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a nonnegative matrix, written
:
is a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
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 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
In matrix theory, the Perron–Frobenius theorem, proved by and , asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector can be chosen to have strictly positive component ...
.
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 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.
See also
*
Metzler matrix In mathematics, a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative (equal to or greater than zero):
: \forall_\, x_ \geq 0.
It is named after the American economist Lloyd Metzler.
Metzler matrices appear in st ...
Bibliography
# Abraham Berman,
Robert J. Plemmons, ''Nonnegative Matrices in the Mathematical Sciences'', 1994, SIAM. .
#A. Berman and R. J. Plemmons, ''Nonnegative Matrices in the Mathematical Sciences'', Academic Press, 1979 (chapter 2),
#R.A. Horn and C.R. Johnson, ''Matrix Analysis'', Cambridge University Press, 1990 (chapter 8).
#
#
# Henryk Minc, ''Nonnegative matrices'', John Wiley&Sons, New York, 1988,
# 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)
#
Richard S. Varga 2002 ''Matrix Iterative Analysis'', Second ed. (of 1962 Prentice Hall edition), Springer-Verlag.
{{Matrix classes
Matrices