Inmathematics, aMetzler matrix is amatrix in which all the off-diagonal components are nonnegative (equal to or greater than zero):

${\displaystyle {\mathrm{\forall }}_{i\ne j}{x}_{ij}\ge 0.}$

It is named after the American economistLloyd Metzler.

Metzler matrices appear in stability analysis oftime delayed differential equations and positivelinear dynamical systems. Their properties can be derived by applying the properties of nonnegative matrices to matrices of the formM + aI, whereM is a Metzler matrix.

Inmathematics, especiallylinear algebra, amatrix is calledMetzler,quasipositive (orquasi-positive) oressentially nonnegative if all of its elements arenon-negative except for those on the main diagonal, which are unconstrained. That is, a Metzler matrix is any matrixA which satisfies

${\displaystyle A=(a_{ij});a_{ij}\ge 0,i\ne j.}$

Metzler matrices are also sometimes referred to as${\displaystyle Z^{(-)}}$-matrices, as aZ-matrix is equivalent to a negated quasipositive matrix.

Theexponential of a Metzler (or quasipositive) matrix is anonnegative matrix because of the corresponding property for the exponential of a nonnegative matrix. This is natural, once one observes that the generator matrices ofcontinuous-time Markov chains are always Metzler matrices, and that probability distributions are always non-negative.

A Metzler matrix has aneigenvector in the nonnegativeorthant because of the corresponding property for nonnegative matrices.

• Perron–Frobenius theorem

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