In mathematics, a nonnegative matrix, written :

\mathbf \geq 0,

is a

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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 number, 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 ( ...

, 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 ...

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 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 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 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_= ...

;

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.

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

Properties

Inversion

Specializations

See also

Bibliography