Relations between Perron—Frobenius results for matrix pencils

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

• Book ID: 104156433
• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 593 KB
• Volume: 287
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(98)10085-x

✦ Synopsis

Two different generalizations of the Perron-Frobenius theory to the matrix pencil Ax = ABx are discussed, and their relationships are studied. In one generalization, which was motivated by economics, the main assumption is that (B -A)-~A is nonnegative. In the second generalization, the main assumption is that there exists a matrix X/> 0 such that A = BX. The equivalence of these two assumptions when B is nonsingular is considered. For p (IB-tAI) < 1, a complete characterization, involving a condition on the digraph of B-IA, is proved. It is conjectured that the characterization holds for p(B-IA) < 1, and partial results are given for this case.