# Article

Keywords:
nonnegative matrices; M-matrices; determinants
Summary:
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.
References:
 S. Ambikkumar and S. W. Drury: Some remarks on a conjecture of Boyle and Handelman. Lin. Alg. Appl. 264 (1997), 63–99. DOI 10.1016/S0024-3795(96)00402-8 | MR 1465857
 J. Ashley.: On the Perron-Frobenius eigenvector for nonnegative integral matrices whose largest eigenvalue is integral. Lin. Alg. Appl. 94 (1987), 103–108. DOI 10.1016/0024-3795(87)90081-4 | MR 0902070 | Zbl 0622.15012
 A. Berman and R. J. Plemmons: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia, 1994. MR 1298430
 M. Boyle and D. Handelman: The spectra of nonnegative matrices via symbolic dynamics. Annals of Math. 133 (1991), 249–316. DOI 10.2307/2944339 | MR 1097240
 M. Fiedler: Untitled private communication. 1982.
 J. Keilson and G. Styan: Markov chains and M-matrices: Inequalities and equalities. J.  Math. Anal. Appl. 41 (1973), 439–459. DOI 10.1016/0022-247X(73)90219-9 | MR 0314873
 I. Koltracht, M. Neumann and D. Xiao: On a question of Boyle and Handelman concerning eigenvalues of nonnegative matrices. Lin. Multilin. Alg. 36 (1993), 125–140. DOI 10.1080/03081089308818282 | MR 1308915