A primer of Perron–Frobenius theory for matrix polynomials
نویسندگان
چکیده
We present an extension of Perron–Frobenius theory to the spectra and numerical ranges of Perron polynomials, namely, matrix polynomials of the form L(λ) = Iλ − Am−1λm−1 − · · · − A1λ− A0, where the coefficient matrices are entrywise nonnegative. Our approach relies on the companion matrix linearization. First, we recount the generalization of the Perron–Frobenius Theorem to Perron polynomials and report some of its consequences. Subsequently, we examine the role of L(λ) in multistep difference equations and provide a multistep version of the Fundamental Theorem of Demography. Finally, we extend Issos’ results on the numerical range of nonnegative matrices to Perron polynomials. © 2004 Elsevier Inc. All rights reserved. AMS classification: 15A48; 15A18; 15A60; 05C50; 39A05; 91B62; 92D25
منابع مشابه
Some results on the block numerical range
The main results of this paper are generalizations of classical results from the numerical range to the block numerical range. A different and simpler proof for the Perron-Frobenius theory on the block numerical range of an irreducible nonnegative matrix is given. In addition, the Wielandt's lemma and the Ky Fan's theorem on the block numerical range are extended.
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