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Xiongping Dai

Publications and source records attributed to Xiongping Dai.

45 records · Page 3Linked to original sources
arXiv2011

Finiteness Property of a Bounded Set of Matrices with Uniformly Sub-Peripheral Spectrum

In the paper, a simple condition guaranteing the finiteness property for a bounded set of matrices is presented. Given a bounded set S of real or complex matrices, it is shown that existence of a sequence of matrix products such that the spectrum of each matrix in this sequence is uniformly sub-peripheral and tends to the joint spectral radius of S, guarantees the spectral finiteness property for S.

math.FA↗
arXiv2011

Extremal ergodic measures and the finiteness property of matrix semigroups

Let $\bS=\{S_1,...,S_K\}$ be a finite set of complex $d\times d$ matrices and $\varSigma_{K}^+$ the compact space of all one-sided infinite sequences $i_{\bcdot}\colon\mathbb{N}\rightarrow\{1,...,K\}$. An ergodic probability $μ_*$ of the Markov shift $θ\colon\varSigma_{K}^+\rightarrow\varSigma_{K}^+;\ i_{\bcdot}\mapsto i_{\bcdot+1}$, is called "extremal" for $\bS$, if $ρ(\bS)=\lim_{n\to\infty}\sqrt[n]{\norm{S_{i_1}...S_{i_n}}}$ holds for $μ_*$-a.e. $i_{\bcdot}\in\varSigma_{K}^+$, where $ρ(\bS)$ denotes the generalized/joint spectral radius of $\bS$. Using extremal norm and Kingman subadditive ergodic theorem, it is shown that $\bS$ has the spectral finiteness property (i.e. $ρ(\bS)=\sqrt[n]{ρ(S_{i_1}...S_{i_n})}$ for some finite-length word $(i_1,...,i_n)$) if and only if for some extremal measure $μ_*$ of $\bS$, it has at least one periodic density point $i_{\bcdot}\in\varSigma_{K}^+$.

math.DS↗
arXiv2011

A Gel'fand-type spectral radius formula and stability of linear constrained switching systems

Using ergodic theory, in this paper we present a Gel'fand-type spectral radius formula which states that the joint spectral radius is equal to the generalized spectral radius for a matrix multiplicative semigroup $\bS^+$ restricted to a subset that need not carry the algebraic structure of $\bS^+$. This generalizes the Berger-Wang formula. Using it as a tool, we study the absolute exponential stability of a linear switched system driven by a compact subshift of the one-sided Markov shift associated to $\bS$.

math.OC↗
arXiv2011

The finite-step realizability of the joint spectral radius of a pair of $d\times d$ matrices one of which being rank-one

We study the finite-step realizability of the joint/generalized spectral radius of a pair of real $d\times d$ matrices, one of which has rank 1. Then we prove that there always exists a finite-length word for which there holds the spectral finiteness property for the set of matrices under consideration. This implies that stability is algorithmically decidable in our case.

math.OC↗