Nonnegative matrix

Not to be confused with Totally positive matrix and Positive-definite matrix.

In mathematics, a nonnegative matrix, written

$\mathbf {X} \geq 0,$

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

$x_{ij}\geq 0\qquad \forall {i,j}.$

A positive matrix is a matrix in which all the elements are strictly greater than zero. The set of positive matrices is the interior of the set of all non-negative matrices. While such matrices are commonly found, the term "positive matrix" 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.

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 n > 1.

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

Bibliography

1. Abraham Berman, Robert J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, 1994, SIAM. ISBN 0-89871-321-8.
2. A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, 1979 (chapter 2), ISBN 0-12-092250-9
3. R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 1990 (chapter 8).
4. Krasnosel'skii, M. A. (1964). Positive Solutions of Operator Equations. Groningen: P.Noordhoff Ltd. pp. 381 pp.
5. Krasnosel'skii, M. A.; Lifshits, Je.A.; Sobolev, A.V. (1990). Positive Linear Systems: The method of positive operators. Sigma Series in Applied Mathematics. Vol. 5. Berlin: Helderman Verlag. pp. 354 pp.
6. Henryk Minc, Nonnegative matrices, John Wiley&Sons, New York, 1988, ISBN 0-471-83966-3
7. 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
8. Richard S. Varga 2002 Matrix Iterative Analysis, Second ed. (of 1962 Prentice Hall edition), Springer-Verlag.