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

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• 作者：Ercan Altınışık^a ; ^{ealtinisik@gazi.edu.tr" class="auth_mail" title="E-mail the corresponding author} ; Ali Keskin^a ; ^{akeskin1729@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Mehmet Yıldız^a ; ^{yildizm78@mynet.com" class="auth_mail" title="E-mail the corresponding author} ; Murat Demirbü ; ken^b ; ^{murdem91@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A18 ; 15A23 ; 15B36 ; 15B48 ; 11B39 ; 11C20
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：493
• 期：Complete
• 页码：1-13
• 全文大小：331 K

文摘

Let K_n$K_n$ be the set of all n×n$n\times n$ lower triangular (0,1)$(0,1)$-matrices with each diagonal element equal to 1, L_n={YY^T:Y∈K_n}$L_n=\{Y{Y}^T:Y\in K_n\}$ and let c_n$c_n$ be the minimum of the smallest eigenvalue of YY^T$Y{Y}^T$ as Y   goes through K_n$K_n$. The Ilmonen–Haukkanen–Merikoski conjecture (the IHM conjecture) states that c_n$c_n$ is equal to the smallest eigenvalue of $Y_0{Y}_0^T$, where Y₀∈K_n$Y_0\in K_n$ with ${(Y_0)}_{ij}=\frac{1-{(-1)}^{i+j}}{2}$ for i>j$i>j$. In this paper, we present a proof of this conjecture. In our proof we use an inequality for spectral radii of nonnegative matrices.