Positive matrix

This article deals with positive matrices, not to be confused with positive definite matrices and totally positive matrices .

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 : ${\displaystyle \mathbf {X} =(x_{ij})_{i,j}}$

${\displaystyle \mathbf {X} \geq 0\Longleftrightarrow \quad \forall {i,j}\quad x_{ij}\geq 0.}$

It is called positive if all of its entries are positive :

${\displaystyle \mathbf {X} >0\Longleftrightarrow \quad \forall {i,j}\quad x_{ij}>0.}$

Applications

• The transition matrix of a Markov chain is a nonnegative matrix.
• The adjacency matrix of a graph is a nonnegative matrix.

Eigenvalues

→ Main article : Perron-Frobenius theorem

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

${\displaystyle \mathbf {X} ={\begin{pmatrix}1&2\\1&1\end{pmatrix}}}$

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 ${\displaystyle 1\pm {\sqrt {2}}}$

${\displaystyle \mathbf {X} ={\begin{pmatrix}2&-1\\-1&2\end{pmatrix}}}$

with the eigenvalues and shows. ${\displaystyle 1}$ ${\displaystyle 3}$

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)