An improvement of an inequality of Fiedler leading to a new conjecture on nonnegative matrices

Goldberger, Assaf ; Neumann, Michael

Czechoslovak Mathematical Journal, Tome 54 (2004) no. 3, pp. 773-780.

Résumé

Suppose that $A$ is an $n\times n$ nonnegative matrix whose eigenvalues are $\lambda = \rho (A), \lambda _2,\ldots , \lambda _n$. Fiedler and others have shown that $\det (\lambda I - A) \le \lambda ^n - \rho ^n$, for all $\lambda > \rho $, with equality for any such $\lambda $ if and only if $A$ is the simple cycle matrix. Let $a_i$ be the signed sum of the determinants of the principal submatrices of $A$ of order $i\times i$, $i = 1,\ldots ,n - 1$. We use similar techniques to Fiedler to show that Fiedler’s inequality can be strengthened to: $\det (\lambda I - A) + \sum _{i = 1}^{n - 1} \rho ^{n - 2i}|a_i|(\lambda - \rho )^i \le \lambda ^n -\rho ^n$, for all $\lambda \ge \rho $. We use this inequality to derive the inequality that: $\prod _{2}^{n}(\rho - \lambda _i) \le \rho ^{n - 2}\sum _{i = 2}^{n}(\rho - \lambda _i)$. In the spirit of a celebrated conjecture due to Boyle-Handelman, this inequality inspires us to conjecture the following inequality on the nonzero eigenvalues of $A$: If $\lambda _1 = \rho (A),\lambda _2,\ldots , \lambda _k$ are (all) the nonzero eigenvalues of $A$, then $\prod _{2}^{k}(\rho - \lambda _i) \le \rho ^{k-2}\sum _{i = 2}^{k}(\rho -\lambda )$. We prove this conjecture for the case when the spectrum of $A$ is real.

Détail

Classification : 15A15, 15A48
Keywords: nonnegative matrices; M-matrices; determinants