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arXiv · 0810.2856

On accuracy of approximation of the spectral radius by the Gelfand formula

Victor Kozyakin
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Abstract

The famous Gelfand formula $ρ(A)= \limsup_{n\to\infty}\|A^{n}\|^{1/n}$ for the spectral radius of a matrix is of great importance in various mathematical constructions. Unfortunately, the range of applicability of this formula is substantially restricted by a lack of estimates for the rate of convergence of the quantities $\|A^{n}\|^{1/n}$ to $ρ(A)$. In the paper this deficiency is made up to some extent. By using the Bochi inequalities we establish explicit computable estimates for the rate of convergence of the quantities $\|A^{n}\|^{1/n}$ to $ρ(A)$. The obtained estimates are then extended for evaluation of the joint spectral radius of matrix sets.

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Victor Kozyakin. 2009-06-30. On accuracy of approximation of the spectral radius by the Gelfand formula. https://doi.org/10.1016/j.laa.2009.07.008

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