Nonnegative matrix explained

In mathematics, a nonnegative matrix, written

is a matrix 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 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 .

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

• Book: Berman, Abraham . Robert J. . Plemmons . Robert J. Plemmons . Nonnegative Matrices in the Mathematical Sciences . SIAM . 1994 . 0-89871-321-8 . 10.1137/1.9781611971262.
  • Book: Horn, R.A. . C.R. . Johnson . 8. Positive and nonnegative matrices . Matrix Analysis . Cambridge University Press . 2nd . 2013 . 978-1-139-78203-6 . 817562427 .
• Book: Krasnosel'skii , M. A. . Mark Krasnosel'skii

. Mark Krasnosel'skii . Positive Solutions of Operator Equations . P. Noordhoff . . 1964 . 609079647.

• Book: Krasnosel'skii . M. A. . Mark Krasnosel'skii . Lifshits . Je.A. . Sobolev . A.V. . Positive Linear Systems: The method of positive operators . Sigma Series in Applied Mathematics . 5 . Helderman Verlag . 3-88538-405-1 . 1409010096 . 1990.
• Book: Minc, Henryk . Nonnegative matrices . Wiley . 1988 . 0-471-83966-3 . 1150971811.
• Book: Seneta, E. . Eugene Seneta . Non-negative matrices and Markov chains . Springer . Springer Series in Statistics . 2nd . 1981 . 978-0-387-29765-1 . 209916821 . 10.1007/0-387-32792-4.
• Book: Varga, R.S. . Richard S. Varga . Nonnegative Matrices . https://link.springer.com/chapter/10.1007/978-3-642-05156-2_2 . 10.1007/978-3-642-05156-2_2 . Matrix Iterative Analysis . Springer . Springer Series in Computational Mathematics . 27 . 2009 . 978-3-642-05156-2 . 31–62 .