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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006977&_mathId=si1.gif&_user=111111111&_pii=S0024379515006977&_rdoc=1&_issn=00243795&md5=6319ab52f6ac1abf217046b2028341fa" title="Click to view the MathML source">K_nclass="mathContainer hidden">class="mathCode">$K_n$ be the set of all class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006977&_mathId=si2.gif&_user=111111111&_pii=S0024379515006977&_rdoc=1&_issn=00243795&md5=599d62a0ea47ceb0af25f9754305a00e" title="Click to view the MathML source">n×nclass="mathContainer hidden">class="mathCode">$n\times n$ lower triangular class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006977&_mathId=si3.gif&_user=111111111&_pii=S0024379515006977&_rdoc=1&_issn=00243795&md5=d317f80a3efabe7f5463e03716f2ad99" title="Click to view the MathML source">(0,1)class="mathContainer hidden">class="mathCode">$(0,1)$-matrices with each diagonal element equal to 1, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006977&_mathId=si18.gif&_user=111111111&_pii=S0024379515006977&_rdoc=1&_issn=00243795&md5=07119bcf21e2e09edee39aaabc746122" title="Click to view the MathML source">L_n={YY^T:Y∈K_n}class="mathContainer hidden">class="mathCode">$L_n=\{Y{Y}^T:Y\in K_n\}$ and let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006977&_mathId=si28.gif&_user=111111111&_pii=S0024379515006977&_rdoc=1&_issn=00243795&md5=b7b75f5a59cae8efdcca5609221c65e3" title="Click to view the MathML source">c_nclass="mathContainer hidden">class="mathCode">$c_n$ be the minimum of the smallest eigenvalue of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006977&_mathId=si178.gif&_user=111111111&_pii=S0024379515006977&_rdoc=1&_issn=00243795&md5=772c78dd3e134bae2f10dd4f18f3f16e" title="Click to view the MathML source">YY^Tclass="mathContainer hidden">class="mathCode">$Y{Y}^T$ as Y   goes through class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006977&_mathId=si1.gif&_user=111111111&_pii=S0024379515006977&_rdoc=1&_issn=00243795&md5=6319ab52f6ac1abf217046b2028341fa" title="Click to view the MathML source">K_nclass="mathContainer hidden">class="mathCode">$K_n$. The Ilmonen–Haukkanen–Merikoski conjecture (the IHM conjecture) states that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006977&_mathId=si28.gif&_user=111111111&_pii=S0024379515006977&_rdoc=1&_issn=00243795&md5=b7b75f5a59cae8efdcca5609221c65e3" title="Click to view the MathML source">c_nclass="mathContainer hidden">class="mathCode">$c_n$ is equal to the smallest eigenvalue of class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006977&_mathId=si29.gif&_user=111111111&_pii=S0024379515006977&_rdoc=1&_issn=00243795&md5=a8c327b412a6361e4135f905f0b04a5e">class="imgLazyJSB inlineImage" height="18" width="41" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379515006977-si29.gif">class="mathContainer hidden">class="mathCode">$Y_0{Y}_0^T$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006977&_mathId=si8.gif&_user=111111111&_pii=S0024379515006977&_rdoc=1&_issn=00243795&md5=ec946e4b0eb932b675b90ae19dd831f7" title="Click to view the MathML source">Y₀∈K_nclass="mathContainer hidden">class="mathCode">$Y_0\in K_n$ with class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006977&_mathId=si9.gif&_user=111111111&_pii=S0024379515006977&_rdoc=1&_issn=00243795&md5=5bc8564ac4df0e788023bcbecf62e33e">class="imgLazyJSB inlineImage" height="25" width="127" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379515006977-si9.gif">class="mathContainer hidden">class="mathCode">${(Y_0)}_{ij}=\frac{1-{(-1)}^{i+j}}{2}$ for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006977&_mathId=si10.gif&_user=111111111&_pii=S0024379515006977&_rdoc=1&_issn=00243795&md5=5fbebc81710b049013dc0966f2bfa15a" title="Click to view the MathML source">i>jclass="mathContainer hidden">class="mathCode">$i>j$. In this paper, we present a proof of this conjecture. In our proof we use an inequality for spectral radii of nonnegative matrices.