Positive matrix
In mathematics , positive matrices and nonnegative matrices occur especially in probability theory , for example for describing Markov chains , and in graph theory .
definition
A matrix is called nonnegative if all of its entries are nonnegative :
It is called positive if all of its entries are positive :
Applications
- The transition matrix of a Markov chain is a nonnegative matrix.
- The adjacency matrix of a graph is a nonnegative matrix.
Eigenvalues
It follows from Perron-Frobenius' theorem that a positive matrix must have a positive eigenvalue . In contrast to totally positive matrices , not all eigenvalues have to be positive.
Examples
Every totally positive matrix is positive, but a positive matrix does not have to be totally positive. For example is the matrix
positive, but not totally positive: the determinant is negative, the eigenvalues are . The same example shows that a positive matrix does not have to be positive definite. Conversely, a positive definite matrix does not have to be positive, like the example
with the eigenvalues and shows.
literature
- Meyer, Carl: Matrix analysis and applied linear algebra. With 1 CD-ROM (Windows, Macintosh and UNIX) and a solutions manual. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. ISBN 0-89871-454-0 pdf (Chapter 8.2)