V.Mehrmann; D.D. Olesky; T.X.T. Phan; P. van den Driessche : Relations between Perron-Frobenius results for matrix pencils

Author(s) :

V.Mehrmann; D.D. Olesky; T.X.T. Phan; P. van den Driessche

Title :

Relations between Perron-Frobenius results for matrix pencils

Electronic source:

application/pdf

Preprint series

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 97-20, 1997

Mathematics Subject Classification :

15A48 [ Positive matrices and their generalizations ]

15A22 [ Matrix pencils ]

05C50 [ Graphs and matrices ]

15A18 [ Eigenvalues of matrices, etc. ]

Abstract :

Two different generalization of Perron-Frobenius theory to the matrix pencil $Ax=\lambda Bx$ are discussed, and their relationships are studied. In one generalization, which was motivated by economics, the main assumption is that $(B - A)^(-1)A$ is nonnegative. In the second generalization, the main assumption is that there exists a matrix $X\le 0$ such that $A=BX$. The equivalence of these two assumptions when B is nonsigular is considered. For $\rho(|B^(-1)A|)<1$, a complete characterization, involving a condition on the digraph of $B^(-1)A$, is proved. It is conjectured that the characterization holds for $\rho(B^(-1)A)<1$, and partial results are given for this case.

Keywords :

nonnegative matrix, generalized eigenvalue problem, digraph, spectral radius

Language :

english

Publication time :

10/1997