Primitive matrices over polynomial semirings
The authors investigate the behavior of powers of matrices over a commutative semiring in which a subset is considered as the `nonnegative' elements that in turn contains a subset as the `positive' elements. They extend the notion of primitivity of a real nonnegative matrix to matrices with univariate polynomial entries and give an analogue of the classical characterization of real nonnegative primitive matrices by irreducibility and aperiodicity for matrices with univariate polynomial entries. They characterize univariate polynomials with nonnegative coefficients which admit a power with strictly positive coefficients and provide a primitivity criterion based on almost linear periodic matrices over dioids. They present a formal treatment of the characterization of primitive real matrices in the vein of \textit{V. Pták} [Czech. Math. J. 8(83), 487--495 (1958; Zbl 0082.24402)], and \textit{J. C. Holladay} and \textit{R. S. Varga} [Proc. Am. Math. Soc. 9, 631--634 (1958; Zbl 0096.00805)].
scientific article; zbMATH DE number 6034478
scientific article