A primer of Perron–Frobenius theory for matrix polynomials. (English) Zbl 1063.15019
The authors present an extension of the Perron-Frobenius theory to spectra and numerical ranges of Perron polynomials. Their approach relies on a companion matrix linearization. First, they recount the generalization of the Perron-Frobenius Theorem to Perron polynomials and report some of its consequences. Subsequently, they examine the role of the matrix polynomial in the multistep version of difference equations and provide a multistep version of the fundamental theorem of demography. Finally, they extend some of J. N. Issos’ results [The field of values of non-negative irreducible matrices. Ph. D. Thesis, Auburn Univ. (1966)] on the numerical range of nonnegative matrices to Perron polynomials.
Reviewer: Grosio Stanilov (Sofia)
MSC:
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 15A54 | Matrices over function rings in one or more variables |
Keywords:
matrix polynomial; nonnegative matrix; companion matrix linearization; Perron-Frobenius theory; Perron polynomial; multistep difference equation; numerical range; spectrum; spectral radiusReferences:
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