On a conjecture of Ilmonen, Haukkanen and Merikoski concerning the smallest eigenvalues of certain GCD related matrices

Let Kn be the set of all n×n lower triangular (0,1)-matrices with each diagonal element equal to 1, Ln={YYT:Y∈Kn} and let cn be the minimum of the smallest eigenvalue of YYT as Y goes through Kn. The Ilmonen–Haukkanen–Merikoski conjecture (the IHM conjecture) states that cn is equal to the smallest...

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Bibliographic Details
Published in:Linear algebra and its applications 2016-03, Vol.493, p.1-13
Main Authors: Altınışık, Ercan, Keskin, Ali, Yıldız, Mehmet, Demirbüken, Murat
Format: Article
Language:English
Subjects:
[formula omitted]-matrix
Eigenvalue
Fibonacci number
GCD matrix
Positive matrix
Spectral radius
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Summary:Let Kn be the set of all n×n lower triangular (0,1)-matrices with each diagonal element equal to 1, Ln={YYT:Y∈Kn} and let cn be the minimum of the smallest eigenvalue of YYT as Y goes through Kn. The Ilmonen–Haukkanen–Merikoski conjecture (the IHM conjecture) states that cn is equal to the smallest eigenvalue of Y0Y0T, where Y0∈Kn with (Y0)ij=1−(−1)i+j2 for i>j. In this paper, we present a proof of this conjecture. In our proof we use an inequality for spectral radii of nonnegative matrices.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2015.11.023