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On the p-norm joint spectral radius.

Jia-li Zhou1

  • 1Department of Mathematics, Zhejiang University, Hangzhou 310027, China. zhoujiali@vip.163.com

Journal of Zhejiang University. Science
|October 21, 2003
PubMed
Summary
This summary is machine-generated.

This study explores the p-norm joint spectral radius for matrices. New formulas are derived, simplifying calculations and offering a new proof for a known spectral radius relation.

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Area of Science:

  • Linear Algebra
  • Matrix Theory
  • Functional Analysis

Background:

  • The joint spectral radius is a key concept in systems theory and control.
  • Understanding spectral radii is crucial for analyzing matrix properties and system stability.
  • Existing literature provides definitions and some properties, but further investigation into specific norms is needed.

Purpose of the Study:

  • To investigate the p-norm joint spectral radius for integer values of p.
  • To derive fundamental formulas for calculating these spectral radii.
  • To provide a simplified proof of the Berger-Wang relation for the infinity-norm joint spectral radius.

Main Methods:

  • Definition of the p-norm joint spectral radius for a bounded collection of square matrices.
  • Development of novel mathematical techniques to derive formulas for integer p.
  • Application of the developed methods to re-prove the Berger-Wang relation.

Main Results:

  • Establishment of basic formulas for the p-norm joint spectral radius.
  • A simplified and accessible proof of the Berger-Wang relation for the infinity-norm.
  • Demonstration of the utility of the new method for spectral radius analysis.

Conclusions:

  • The derived formulas offer efficient tools for computing p-norm joint spectral radii.
  • The simplified proof enhances the understanding of the Berger-Wang relation.
  • This work contributes to the theoretical framework of joint spectral radius analysis in linear algebra.