Metzler matrix

A Metzler matrix is a matrix whose elements outside of the main diagonal all have non-negative values.

${\displaystyle \qquad \forall _{i\neq j}\,x_{ij}\geq 0.}$

The namesake of these matrices is the American economist Lloyd Metzler . Other names are quasi-positive matrix or substantially non-negative matrix .

Metzler matrices appear in the stability analysis of retarded differential equations and in positive linear dynamic systems .

Definition and terminology

A Metzler matrix fulfills the condition ${\displaystyle A}$

${\displaystyle A=(a_{ij});\quad a_{ij}\geq 0,\quad i\neq j.}$

properties

The matrix exponential of a Metzler matrix is ​​a nonnegative matrix . This can be illustrated in such a way that the generating matrices of a time-continuous Markov process are always Metzler matrices and probability distributions are always positive.

A Metzler matrix has at least one eigenvector in the orthant ${\displaystyle \Omega =\left\{(x_{1},x_{2},\ldots ,x_{d})\in \mathbb {R} ^{d}\mid \forall i=1,\ldots ,d\colon \quad 0\leq x_{i}\right\}.}$

Relevant sentences

• Perron-Frobenius' theorem