Hankel tensors, Vandermonde tensors and their positivities

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• 作者：Changqing Xu ^{cqxurichard@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：53A45 ; 15A69
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 February 2016
• 年：2016
• 卷：491
• 期：Complete
• 页码：56-72
• 全文大小：374 K

文摘

An mth order n  -dimensional Hankel tensor is defined as a tensor class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515001184&_mathId=si1.gif&_user=111111111&_pii=S0024379515001184&_rdoc=1&_issn=00243795&md5=0bac7376db833c4c47d7a43db6397976" title="Click to view the MathML source">Aclass="mathContainer hidden">class="mathCode">$\mathcal{A}$ satisfying class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515001184&_mathId=si2.gif&_user=111111111&_pii=S0024379515001184&_rdoc=1&_issn=00243795&md5=897ffe95de63d287802f8ec32751e3a4" title="Click to view the MathML source">A_i1…im≡A_{i1+⋯+im−m}class="mathContainer hidden">class="mathCode">$A_{i_1\dots i_m}\equiv A_{i_1+\cdots +i_m-m}$ for some numbers class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515001184&_mathId=si3.gif&_user=111111111&_pii=S0024379515001184&_rdoc=1&_issn=00243795&md5=1415b840310927a6484417ed23325b42" title="Click to view the MathML source">A₀,A₁,…,A_m(n−1)class="mathContainer hidden">class="mathCode">$A_0,A_1,\dots ,A_{m(n-1)}$. A Hankel tensor possesses a Vandermonde decomposition (VD) class="mathmlsrc">class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515001184&_mathId=si4.gif&_user=111111111&_pii=S0024379515001184&_rdoc=1&_issn=00243795&md5=ac3575c7862d4e49849880847fd151c1">class="imgLazyJSB inlineImage" height="35" width="102" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379515001184-si4.gif">class="mathContainer hidden">class="mathCode">$\mathcal{A}=\sum _{k=1}^r{\lambda }_k{\mathbf{u}}_k^m$ where class="mathmlsrc">class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515001184&_mathId=si5.gif&_user=111111111&_pii=S0024379515001184&_rdoc=1&_issn=00243795&md5=19d7e1cafd36a6f86fd5ad0abef46bf6">class="imgLazyJSB inlineImage" height="20" width="198" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379515001184-si5.gif">class="mathContainer hidden">class="mathCode">${\mathbf{u}}_k={(1,w_k,w_k^2,\dots ,w_k^{n-1})}^T$ is called a Vandermonde vector (V-vector). class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515001184&_mathId=si1.gif&_user=111111111&_pii=S0024379515001184&_rdoc=1&_issn=00243795&md5=0bac7376db833c4c47d7a43db6397976" title="Click to view the MathML source">Aclass="mathContainer hidden">class="mathCode">$\mathcal{A}$ is called a Vandermonde tensor (V-tensor) if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515001184&_mathId=si1.gif&_user=111111111&_pii=S0024379515001184&_rdoc=1&_issn=00243795&md5=0bac7376db833c4c47d7a43db6397976" title="Click to view the MathML source">Aclass="mathContainer hidden">class="mathCode">$\mathcal{A}$ has a VD with each class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515001184&_mathId=si7.gif&_user=111111111&_pii=S0024379515001184&_rdoc=1&_issn=00243795&md5=008795b23eb4ac403d63e204aef6befb" title="Click to view the MathML source">λ_k=1class="mathContainer hidden">class="mathCode">${\lambda }_k=1$. V-tensors are the natural extension of Vandermonde matrices. It is easy to see that all even order V-tensors are positive semidefinite (psd) and thus copositive. An odd order real symmetric tensor is psd only if it is zero. The problem when an odd order Hankel tensor is copositive is open. We present a necessary and sufficient condition for a rank-2 odd-order symmetric real tensor to be copositive. Some necessary conditions for a general V-tensor to be copositive are also presented. The singularity of V-tensors is also investigated, and we show that a V-tensor class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515001184&_mathId=si1.gif&_user=111111111&_pii=S0024379515001184&_rdoc=1&_issn=00243795&md5=0bac7376db833c4c47d7a43db6397976" title="Click to view the MathML source">Aclass="mathContainer hidden">class="mathCode">$\mathcal{A}$ is singular if its V-rank is less than its dimension. This condition becomes necessary if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515001184&_mathId=si1.gif&_user=111111111&_pii=S0024379515001184&_rdoc=1&_issn=00243795&md5=0bac7376db833c4c47d7a43db6397976" title="Click to view the MathML source">Aclass="mathContainer hidden">class="mathCode">$\mathcal{A}$ is of odd order.