Metzler matrix

In mathematics, a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative (equal to or greater than zero):

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

It is named after the American economist Lloyd Metzler.

Metzler matrices appear in stability analysis of time delayed differential equations and positive linear dynamical systems. Their properties can be derived by applying the properties of nonnegative matrices to matrices of the form M + aI, where M is a Metzler matrix.