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A “dynamic” proof of the Frobenius-Perron theorem for Metzler matrices. (English) Zbl 0696.15015

Probability, statistics, and mathematics, Pap. in Honor of Samuel Karlin, 17-26 (1989).
[For the entire collection see Zbl 0679.00013.]
The class of matrices with non-negative off-diagonal entries (Metzler matrices) is considered. A linear dynamical system governed by a Metzler matrix has the following property: if the forcing term is nonnegative and the initial condition positive then the solution remains also positive at any time. This statement (Karlin’s theorem) is used to give an alternative proof of Frobenius-Perron theorem for the mentioned class of matrices.
Reviewer: D.Janovska

MSC:

15B48 Positive matrices and their generalizations; cones of matrices
15A18 Eigenvalues, singular values, and eigenvectors
91B60 Trade models
93C15 Control/observation systems governed by ordinary differential equations
93C05 Linear systems in control theory

Citations:

Zbl 0679.00013