In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a nonnegative matrix, written
:
is a
matrix
Matrix (: matrices or matrixes) or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the m ...
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
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 th ...
.
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
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular or sounder, a group of boar, see List of animal names
* Singular (band), a Thai jazz pop duo
*'' 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
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, ''s ...
;
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 () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
non-negative matrix.
See also
*
Metzler matrix
Bibliography
*
*
*
*
*
*
*
*
* Andrzej Cichocki; Rafel Zdunek; Anh Huy Phan; Shun-ichi Amari: ''Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation'', John Wiley & Sons,ISBN 978-0-470-74666-0 (2009).
{{Matrix classes
Matrices (mathematics)