An improvement of an inequality of Fiedler leading to a new conjecture on nonnegative matrices
Czechoslovak Mathematical Journal, Tome 54 (2004) no. 3, pp. 773-780
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.
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.
@article{CMJ_2004_54_3_a18,
author = {Goldberger, Assaf and Neumann, Michael},
title = {An improvement of an inequality of {Fiedler} leading to a new conjecture on nonnegative matrices},
journal = {Czechoslovak Mathematical Journal},
pages = {773--780},
year = {2004},
volume = {54},
number = {3},
mrnumber = {2086733},
zbl = {1080.15502},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMJ_2004_54_3_a18/}
}
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 SP - 773 EP - 780 VL - 54 IS - 3 UR - http://geodesic.mathdoc.fr/item/CMJ_2004_54_3_a18/ LA - en ID - CMJ_2004_54_3_a18 ER -
%0 Journal Article %A Goldberger, Assaf %A Neumann, Michael %T An improvement of an inequality of Fiedler leading to a new conjecture on nonnegative matrices %J Czechoslovak Mathematical Journal %D 2004 %P 773-780 %V 54 %N 3 %U http://geodesic.mathdoc.fr/item/CMJ_2004_54_3_a18/ %G en %F CMJ_2004_54_3_a18
Goldberger, Assaf; Neumann, Michael. An improvement of an inequality of Fiedler leading to a new conjecture on nonnegative matrices. Czechoslovak Mathematical Journal, Tome 54 (2004) no. 3, pp. 773-780. http://geodesic.mathdoc.fr/item/CMJ_2004_54_3_a18/
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