Strong shift equivalence and the Generalized Spectral Conjecture for nonnegative matrices

Mike Boyle; Scott Schmieding

doi:10.1016/j.laa.2015.06.004

Strong shift equivalence and the Generalized Spectral Conjecture for nonnegative matrices

Mike Boyle^*, Scott Schmieding

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

Given matrices A and B shift equivalent over a dense subring R of ℝ, with A primitive, we show that B is strong shift equivalent over R to a primitive matrix. This result shows that the weak form of the Generalized Spectral Conjecture for primitive matrices implies the strong form. The foundation of this work is the recent result that for any ring R, the group NK₁(R) of algebraic K-theory classifies the refinement of shift equivalence by strong shift equivalence for matrices over R.

Original language	English (US)
Pages (from-to)	231-243
Number of pages	13
Journal	Linear Algebra and Its Applications
Volume	498
DOIs	https://doi.org/10.1016/j.laa.2015.06.004
State	Published - Jun 1 2016

Keywords

Nonnegative matrix
Shift equivalence
Spectra
Spectral conjecture

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1016/j.laa.2015.06.004

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abstract = "Given matrices A and B shift equivalent over a dense subring R of ℝ, with A primitive, we show that B is strong shift equivalent over R to a primitive matrix. This result shows that the weak form of the Generalized Spectral Conjecture for primitive matrices implies the strong form. The foundation of this work is the recent result that for any ring R, the group NK1(R) of algebraic K-theory classifies the refinement of shift equivalence by strong shift equivalence for matrices over R.",

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AU - Boyle, Mike

AU - Schmieding, Scott

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N2 - Given matrices A and B shift equivalent over a dense subring R of ℝ, with A primitive, we show that B is strong shift equivalent over R to a primitive matrix. This result shows that the weak form of the Generalized Spectral Conjecture for primitive matrices implies the strong form. The foundation of this work is the recent result that for any ring R, the group NK1(R) of algebraic K-theory classifies the refinement of shift equivalence by strong shift equivalence for matrices over R.

AB - Given matrices A and B shift equivalent over a dense subring R of ℝ, with A primitive, we show that B is strong shift equivalent over R to a primitive matrix. This result shows that the weak form of the Generalized Spectral Conjecture for primitive matrices implies the strong form. The foundation of this work is the recent result that for any ring R, the group NK1(R) of algebraic K-theory classifies the refinement of shift equivalence by strong shift equivalence for matrices over R.

KW - Nonnegative matrix

KW - Shift equivalence

KW - Spectra

KW - Spectral conjecture

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SP - 231

EP - 243

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

Strong shift equivalence and the Generalized Spectral Conjecture for nonnegative matrices

Abstract

Keywords

ASJC Scopus subject areas

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