Strong Shift Equivalence and Positive Doubly Stochastic Matrices
| Autor: | Sompong Chuysurichay |
|---|---|
| Rok vydání: | 2014 |
| Předmět: |
Doubly stochastic matrix
Numerical Analysis Algebra and Number Theory Stochastic matrix Dynamical Systems (math.DS) Upper and lower bounds Spectral line Combinatorics Matrix (mathematics) Bounded function FOS: Mathematics Discrete Mathematics and Combinatorics Primary 15B48 Secondary 37B10 15A21 15B51 Geometry and Topology Nonnegative matrix Matrix analysis Mathematics - Dynamical Systems Mathematics |
| DOI: | 10.48550/arxiv.1407.2485 |
| Popis: | We give sufficient conditions for a positive stochastic matrix to be similar and strong shift equivalent over $\mathbb{R}_+$ to a positive doubly stochastic matrix through matrices of the same size. We also prove that every positive stochastic matrix is strong shift equivalent over $\mathbb{R}_+$ to a positive doubly stochastic matrix. Consequently, the set of nonzero spectra of primitive stochastic matrices over $\mathbb{R}$ with positive trace and the set of nonzero spectra of positive doubly stochastic matrices over $\mathbb{R}$ are identical. We exhibit a class of $2\times 2$ matrices, pairwise strong shift equivalent over $\mathbb R_+$ through $2\times 2$ matrices, for which there is no uniform upper bound on the minimum lag of a strong shift equivalence through matrices of bounded size. In contrast, we show for any $n\times n$ primitive matrix of positive trace that the set of positive $n\times n$ matrices similar to it contains only finitely many SSE-$\mathbb R_+$ classes. Comment: 12 pages |
| Databáze: | OpenAIRE |
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