An Improvement of an Inequality of Fiedler Leading to a New Conjecture on Nonnegative Matrices |
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Authors: | Assaf Goldberger Michael Neumann |
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Institution: | (1) Department of Mathematics, University of Connecticut, Storrs, Connecticut, 06269-3009, USA |
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Abstract: | Suppose that A is an n × n nonnegative matrix whose eigenvalues are = (A), 2, ..., n. Fiedler and others have shown that \det( I -A) n - n, for all > with equality for any such 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 × i, i=1, ..., n - 1. We use similar techniques to Fiedler to show that Fiedler's inequality can be strengthened to:
for all . We use this inequality to derive the inequality that:
. 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 1 = (A), 2,...,k
are (all) the nonzero eigenvalues of A, then
. We prove this conjecture for the case when the spectrum of A is real. |
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Keywords: | nonnegative matrices M-matrices determinants |
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