| Title:
|
An improvement of an inequality of Fiedler leading to a new conjecture on nonnegative matrices (English) |
| Author:
|
Goldberger, Assaf |
| Author:
|
Neumann, Michael |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
54 |
| Issue:
|
3 |
| Year:
|
2004 |
| Pages:
|
773-780 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
nonnegative matrices |
| Keyword:
|
M-matrices |
| Keyword:
|
determinants |
| MSC:
|
15A15 |
| MSC:
|
15A48 |
| idZBL:
|
Zbl 1080.15502 |
| idMR:
|
MR2086733 |
| . |
| Date available:
|
2009-09-24T11:17:29Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127928 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
[3] A. Berman and R. J. Plemmons: Nonnegative Matrices in the Mathematical Sciences.SIAM, Philadelphia, 1994. MR 1298430 |
| Reference:
|
[4] M. Boyle and D. Handelman: The spectra of nonnegative matrices via symbolic dynamics.Annals of Math. 133 (1991), 249–316. MR 1097240, 10.2307/2944339 |
| Reference:
|
[5] M. Fiedler: Untitled private communication.1982. |
| Reference:
|
[6] J. Keilson and G. Styan: Markov chains and M-matrices: Inequalities and equalities.J. Math. Anal. Appl. 41 (1973), 439–459. MR 0314873, 10.1016/0022-247X(73)90219-9 |
| Reference:
|
[7] 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. MR 1308915, 10.1080/03081089308818282 |
| . |
