A lower bound for the spectral radius of complex matrices
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.
scientific article; zbMATH DE number 7613283
scientific article