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 :

\mathbf \geq 0,

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'' (franchis ...

in which all the elements are equal to or greater than zero, that is, :

x_ \geq 0\qquad \forall .

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 In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a c ...

, 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 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

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and every row and column sum/product of a nonnegative matrix is nonnegative.

Inversion

The inverse of any

non-singular In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In cas ...

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-p ...

is a non-negative matrix. If the non-singular M-matrix is also symmetric then it is called a

Stieltjes matrix In mathematics, particularly matrix theory, a Stieltjes matrix, named after Thomas Joannes Stieltjes, is a real symmetric positive definite matrix with nonpositive off-diagonal entries. A Stieltjes matrix is necessarily an M-matrix. Every ''n× ...

. 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 expone ...

: 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 In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It is also called a probability matrix, transition matrix, ...

;

doubly stochastic matrix In mathematics, especially in probability and combinatorics, a doubly stochastic matrix (also called bistochastic matrix) is a square matrix X=(x_) of nonnegative real numbers, each of whose rows and columns sums to 1, i.e., :\sum_i x_=\sum_j x_=1 ...

;

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 definiti ...

non-negative matrix.

Bibliography

# Abraham Berman,

Robert J. Plemmons Robert James Plemmons (born December 18, 1938) is an American mathematician specializing in computational mathematics. He is the Emeritus Z. Smith Reynolds Professor of Mathematics and Computer Science at Wake Forest University. In 1979, Plemmon ...

, ''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 Richard Steven Varga (October 9, 1928 - February 25, 2022) was an American mathematician who specialized in numerical analysis and linear algebra. He was an Emeritus University Professor of Mathematical Sciences at Kent State University and an a ...

2002 ''Matrix Iterative Analysis'', Second ed. (of 1962 Prentice Hall edition), Springer-Verlag. {{Matrix classes Matrices

Properties

Inversion

Specializations

See also

Bibliography