A lower bound for the spectral radius of complex matrices
type: Item
instance of: scholarly article
title: A lower bound for the spectral radius of complex matrices
zbMATH Open document ID: 1506.15009
author: Minghua Lin
author: Jiao Zhang
author: Teng Zhang
published in: Linear Algebra and its Applications
publication date: 2022-11-08T00:00:00Z
review text: The spectral radius of a complex matrix \(A\) is defined as the largest eigenvalue of \(A\) in modulus, i.e., \[ \rho(A)=\max \{|\lambda|: A-\lambda I \text{ is singular}\}. \] Denote the Frobenius norm of \(A\) by \(\|A\|\) and the \(r\)-th compound matrix of \(A\) by \(C_r(A)\). For an arbitrary \(n\times n\) real matrix \(A\), \textit{B. G. Horne} [Linear Algebra Appl. 263, 261--273 (1997; Zbl 0889.15014)] presented a lower bound for the spectral radius based on a classical result of F. Lucas (see, e.g., [\textit{P. Borwein} and \textit{T. Erdélyi}, Polynomials and polynomial inequalities. New York, NY: Springer-Verlag (1995; Zbl 0840.26002), p. 18]). In this paper, the authors use Laguerre's theorem on the polar derivative to derive a new lower bound for the spectral radius for an \(n\times n\) complex matrix \(A\). Theorem. Let \(A\) be an \(n\times n\) complex matrix with {rank}\((A)\geq 2\). Then \[ \rho(A)\geq \max\left\{\left|\frac{-b_1\pm \sqrt{b_1^2-b_0b_2}}{b_2} \right| \right\}, \] where \begin{align*} b_0 &=\sum_{i=0}^{n-2}(-1)^{n-i}(n-i)(n-i-1)\|A\|^i \operatorname{tr}\, C_{n-i}(A),\\ b_1 &= \sum_{i=0}^{n-2}(-1)^{n-i-1}(i+1)(n-i-1)\|A\|^i \operatorname{tr}\, C_{n-i-1}(A), \\ b_2 &= \sum_{i=0}^{n-2}(-1)^{n-i-2}(i+1)(i+2)\|A\|^i \operatorname{tr}\, C_{n-i-2}(A). \end{align*} The authors also compare the lower bound obtained in [\textit{B. G. Horne}, Linear Algebra Appl. 263, 261--273 (1997; Zbl 0889.15014)] with the one obtained in the above theorem and point out that when the underlying matrix is entrywise nonnegative the new lower bound is competitive.
reviewed by: Graça Soares
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 7613283
zbMATH Keywords: spectral radius
zbMATH Keywords: Laguerre's theorem
zbMATH Keywords: polar derivative
MaRDI profile type: Publication
full work available at URL: j.laa.2022.09.012
OpenAlex ID: W4296984454
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DOI: 10.1016/J.LAA.2022.09.012
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description: scientific article; zbMATH DE number 7613283