Metzler matrix

In mathematics, especially linear algebra, a matrix is called Metzler, quasipositive (or quasi-positive) or essentially nonnegative if all of its elements are non-negative except for those on the main diagonal, which are unconstrained. That is, a Metzler matrix is any matrix A which satisfies :A=(a_{ij});\quad a_{ij}\geq 0, \quad i\neq j. Metzler matrices are also sometimes referred to as Z^{(-)}-matrices, as a Z-matrix is equivalent to a negated quasipositive matrix. == Properties ==

The exponential of a Metzler (or quasipositive) matrix is a nonnegative matrix because of the corresponding property for the exponential of a nonnegative matrix. This is natural, once one observes that the generator matrices of continuous-time Markov chains are always Metzler matrices, and that probability distributions are always non-negative. A Metzler matrix has an eigenvector in the nonnegative orthant because of the corresponding property for nonnegative matrices. == Relevant theorems ==

• Perron–Frobenius theorem == See also ==