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

Goldberger, Assaf, and Neumann, Michael. "An improvement of an inequality of Fiedler leading to a new conjecture on nonnegative matrices." Czechoslovak Mathematical Journal 54.3 (2004): 773-780. <http://eudml.org/doc/30899>.

@article{Goldberger2004,
abstract = {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.},
author = {Goldberger, Assaf, Neumann, Michael},
journal = {Czechoslovak Mathematical Journal},
keywords = {nonnegative matrices; M-matrices; determinants; nonnegative matrices; M-matrices; determinants},
language = {eng},
number = {3},
pages = {773-780},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {An improvement of an inequality of Fiedler leading to a new conjecture on nonnegative matrices},
url = {http://eudml.org/doc/30899},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Goldberger, Assaf
AU - Neumann, Michael
TI - An improvement of an inequality of Fiedler leading to a new conjecture on nonnegative matrices
JO - Czechoslovak Mathematical Journal
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 3
SP - 773
EP - 780
AB - 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.
LA - eng
KW - nonnegative matrices; M-matrices; determinants; nonnegative matrices; M-matrices; determinants
UR - http://eudml.org/doc/30899
ER -