Testing the irreducibility of nonsquare Perron-Frobenius systems

Generalized Perron-Frobenius Theorem for Nonsquare Matrices

The celebrated Perron–Frobenius (PF) theorem is stated for irreducible nonnegative square matrices, and provides a simple characterization of their eigenvectors and eigenvalues. The importance of this theorem stems from the fact that eigenvalue problems on such matrices arise in many fields of science and engineering, including dynamical systems theory, economics, statistics and optimization. H...

Generalized Perron-Frobenius Theorem for Multiple Choice Matrices, and Applications

The celebrated Perron–Frobenius (PF) theorem is stated for irreducible nonnegative square matrices, and provides a simple characterization of their eigenvectors and eigenvalues. The importance of this theorem stems from the fact that eigenvalue problems on such matrices arise in many fields of science and engineering, including dynamical systems theory, economics, statistics and optimization. H...

Perron-Frobenius Type Results on the Numerical Range

We present results connecting the shape of the numerical range to intrinsic properties of a matrix A. When A is a nonnegative matrix, these results are to a large extent analogous to the Perron-Frobenius theory, especially as it pertains to irreducibility and cyclicity in the combinatorial sense. Special attention is given to polygonal, circular and elliptic numerical ranges. The main vehicles ...

Compact weighted Frobenius-Perron operators and their spectra

In this note we characterize the compact weighted Frobenius-Perron operator $p$ on $L^1(Sigma)$ and determine their spectra. We also show that every weakly compact weighted Frobenius-Perron operator on $L^1(Sigma)$ is compact.

Spectrum Properties of the Koopman and Frobenius-Perron Operators