Nonnegative matrix
In this article, we will explore in detail Nonnegative matrix, a topic of great relevance in contemporary society. Nonnegative matrix has captured the attention of numerous experts and academics due to its significant impact on various aspects of daily life. Through meticulous analysis and the collection of empirical evidence, we will seek to shed light on the different nuances and facets that characterize Nonnegative matrix, in order to provide a comprehensive and enriching vision on this topic. In addition, the practical and theoretical implications of Nonnegative matrix will be examined, in order to promote a deeper and more detailed understanding of its importance in the current landscape.
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 [clarification needed] 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
Bibliography
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- Berman & Plemmons 1994, 2. Nonnegative Matrices pp. 26–62. doi:10.1137/1.9781611971262.ch2
- Horn, R.A.; Johnson, C.R. (2013). "8. Positive and nonnegative matrices". Matrix Analysis (2nd ed.). Cambridge University Press. ISBN 978-1-139-78203-6. OCLC 817562427.
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- Varga, R.S. (2009). "Nonnegative Matrices". Matrix Iterative Analysis. Springer Series in Computational Mathematics. Vol. 27. Springer. pp. 31–62. doi:10.1007/978-3-642-05156-2_2. ISBN 978-3-642-05156-2.
- 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).
