Matrix roots of imprimitive irreducible nonnegative matrices

Jordan chains of $h$-cyclic matrices

Arising from the classification of the matrix-roots of a nonnegative imprimitive irreducible matrix, we present results concerning the Jordan chains of an $h$-cyclic matrix. We also present ancillary results applicable to nonnegative imprimitive irreducible matrices and demonstrate these results via examples.

具有本原与非负不可约标准部的对偶数矩阵

Dual Number Matrices with Primitive and Irreducible Nonnegative Standard Parts

本文将Perron-Frobenius理论推广到标准部本原且非负不可约的对偶数矩阵，证明其正对偶数特征值及特征向量的存在性，给出对偶部显式公式，并基于Collatz极小极大定理构造算法，用于分析对偶马尔可夫链与扰动马尔可夫链的稳态距离。

The higher rank numerical range of nonnegative matrices

In this article the well known "Perron-Frobenius theory" is investigated involving the higher rank numerical range $Λ_{k}(A)$ of an irreducible and entrywise nonnegative matrix $A$ and extending the notion of elements of maximum modulus in $Λ_{k}(A)$. Further, an application of this theory to the $Λ_{k}(L(λ))$ of a Perron polynomial $L(λ)$ is elaborated via its companion matrix $C_{L}$.

On commuting matrices in max algebra and in classical nonnegative algebra

This paper studies commuting matrices in max algebra and nonnegative linear algebra. Our starting point is the existence of a common eigenvector, which directly leads to max analogues of some classical results for complex matrices. We also investigate Frobenius normal forms of commuting matrices, particularly when the Perron roots of the components are distinct. For the case of max algebra, we show how the intersection of eigencones of commuting matrices can be described, and we consider connections with Boolean algebra which enables us to prove that two commuting irreducible matrices in max a…

Matrix semigroups with constant spectral radius

Multiplicative matrix semigroups with constant spectral radius (c.s.r.) are studied and applied to several problems of algebra, combinatorics, functional equations, and dynamical systems. We show that all such semigroups are characterized by means of irreducible ones. Each irreducible c.s.r. semigroup defines walks on Euclidean sphere, all its nonsingular elements are similar (in the same basis) to orthogonal. We classify all nonnegative c.s.r. semigroups and arbitrary low-dimensional semigroups. For higher dimensions, we describe five classes and leave an open problem on completeness of that …

On the invariant faces associated with a cone-preserving map

For an $n \times n$ nonnegative matrix $P$, an isomorphism is obtained between the lattice of initial subsets (of ${1,...,n}$) for $P$ and the lattice of $P$-invariant faces of the nonnegative orthant $\IR^{n}_{+}$. Motivated by this isomorphism, we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone-preserving map on a polyhedral cone, formulated in terms of its invariant faces. In particular, we obtain the following extension of the famous Rothblum Index Theorem for a nonnegative matrix: If $A$ leave…