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

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
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