In
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
Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra where a matrix is factorized into (usually) two matrices and , with the property that ...
.
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 In mathematics, especially linear algebra, an ''M''-matrix is a ''Z''-matrix with eigenvalues whose real parts are nonnegative. The set of non-singular ''M''-matrices are a subset of the class of ''P''-matrices, and also of the class of inverse ...
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
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expon ...
: 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
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
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