Inverse problems for -symmetric matrices in structural dynamic model updating

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• 作者：Wei-Ru Xu ^{weiruxu@foxmail.com" class="auth_mail" title="E-mail the corresponding author} ; Guo-Liang Chen ; ^{glchen@math.ecnu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Inverse problem ; (R ; S)(R ; S)-symmetric matrix ; Generalized singular value decomposition ; Canonical correlation decomposition ; Projection theorem
• 刊名：Computers & Mathematics with Applications
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：71
• 期：5
• 页码：1074-1088
• 全文大小：469 K

文摘

Let class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si15.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=310533fd14870cd386e5e24e9ef3dde4">class="imgLazyJSB inlineImage" height="15" width="82" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0898122116300220-si15.gif">class="mathContainer hidden">class="mathCode"> be nontrivial involutions, i.e., class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si16.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=87f05b08d13cfa194dfdde11c2781f74" title="Click to view the MathML source">R=R⁻¹≠±I_nclass="mathContainer hidden">class="mathCode">$R=R^{-1}\ne \pm I_n$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si17.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=c128b2fdb651aa1c6cd52d6bb3bd67c4" title="Click to view the MathML source">S=S⁻¹≠±I_nclass="mathContainer hidden">class="mathCode">$S=S^{-1}\ne \pm I_n$. A matrix class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si18.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=ea7993ed11a3757db3d8f67f7611075f" title="Click to view the MathML source">A∈C^n×nclass="mathContainer hidden">class="mathCode">$A\in {\mathbb{C}}^{n\times n}$ is referred to as class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si14.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=2e3ffe6cd983868fd507ef856d6ab293" title="Click to view the MathML source">(R,S)class="mathContainer hidden">class="mathCode">$(R,S)$-symmetric if and only if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si20.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=7290375c6588ce950d22b9dc7f2367bc" title="Click to view the MathML source">RAS=Aclass="mathContainer hidden">class="mathCode">$RAS=A$. The set of all class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si14.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=2e3ffe6cd983868fd507ef856d6ab293" title="Click to view the MathML source">(R,S)class="mathContainer hidden">class="mathCode">$(R,S)$-symmetric matrices of order class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si22.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=dfe1fd6835d792b245735b775991d64d" title="Click to view the MathML source">nclass="mathContainer hidden">class="mathCode">$n$ is denoted by class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si23.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=661bfd82d5f6faa208a6aa2dd9d21fec">class="imgLazyJSB inlineImage" height="17" width="71" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0898122116300220-si23.gif">class="mathContainer hidden">class="mathCode">${\mathcal{C}}_s^{n\times n}(R,S)$. Given a full column rank matrix class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si24.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=081bbef6d8a5f92ca228cf9315c9148b" title="Click to view the MathML source">X∈C^n×mclass="mathContainer hidden">class="mathCode">$X\in {\mathbb{C}}^{n\times m}$, a matrix class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si25.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=c0383ad44dff4360ab1b90bfd04fab9a" title="Click to view the MathML source">B∈C^m×mclass="mathContainer hidden">class="mathCode">$B\in {\mathbb{C}}^{m\times m}$ and a matrix class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si26.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=364e8ad51a84c248db355b15989a983b" title="Click to view the MathML source">A^∗∈C^n×nclass="mathContainer hidden">class="mathCode">$A^{\ast }\in {\mathbb{C}}^{n\times n}$. In structural dynamic model updating, we usually consider the sets class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si27.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=aaccd45cc472c025f5427247311a701c">class="imgLazyJSB inlineImage" height="19" width="262" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0898122116300220-si27.gif">class="mathContainer hidden">class="mathCode"> and class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si28.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=cea330e617c1a5f9968d49e5c067feff">class="imgLazyJSB inlineImage" height="19" width="326" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0898122116300220-si28.gif">class="mathContainer hidden">class="mathCode"> in the Frobenius norm sense, where the superscript class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si29.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=fb36557dc85bedf38547434ae0911b1d" title="Click to view the MathML source">Hclass="mathContainer hidden">class="mathCode">$\text{H}$ denotes conjugate transpose. Then we characterize the unique matrices class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si30.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=e09c938bd16fb3119307a47b1ae8a6cb">class="imgLazyJSB inlineImage" height="30" width="148" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0898122116300220-si30.gif">class="mathContainer hidden">class="mathCode">$\tilde{A}=arg\underset{A\in {\mathcal{S}}_1}{min}\Vert A-A^{\ast }\Vert$ and class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si31.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=d3c20793143eb427fcadf1b0f281e5bd">class="imgLazyJSB inlineImage" height="30" width="149" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0898122116300220-si31.gif">class="mathContainer hidden">class="mathCode">$\widehat{A}=arg\underset{A\in {\mathcal{S}}_2}{min}\Vert A-A^{\ast }\Vert$ when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si32.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=52d398e228513f898563063e1bda6f17" title="Click to view the MathML source">R=R^Hclass="mathContainer hidden">class="mathCode">$R=R^{\text{H}}$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si33.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=91e2d92dac35b04efae7a0df509b82de" title="Click to view the MathML source">S=S^Hclass="mathContainer hidden">class="mathCode">$S=S^{\text{H}}$. By using the generalized singular value decomposition (GSVD), the necessary and sufficient conditions for the non-emptiness of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si34.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=8c009633ee0508ff29b7dc10a44c0be2" title="Click to view the MathML source">S₁class="mathContainer hidden">class="mathCode">${\mathcal{S}}_1$ and the general representations of the elements in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si34.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=8c009633ee0508ff29b7dc10a44c0be2" title="Click to view the MathML source">S₁class="mathContainer hidden">class="mathCode">${\mathcal{S}}_1$ and class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si36.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=6e8b0f71352e76e420ffce32bbfbe8ef">class="imgLazyJSB inlineImage" height="17" width="12" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0898122116300220-si36.gif">class="mathContainer hidden">class="mathCode">$\tilde{A}$ are derived, respectively. The analytical expressions of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si37.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=e5d15d3171a3566810a91fcf3552e249" title="Click to view the MathML source">A∈S₂class="mathContainer hidden">class="mathCode">$A\in {\mathcal{S}}_2$ and class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0898122116300220&_mathId=si38.gif&_user=111111111&_pii=S0898122116300220&_rdoc=1&_issn=08981221&md5=99c844826345c5da92f5d6a4e58fffac">class="imgLazyJSB inlineImage" height="17" width="15" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0898122116300220-si38.gif">class="mathContainer hidden">class="mathCode">$\widehat{A}$ are also obtained by using the GSVD, the canonical correlation decomposition (CCD) and the projection theorem. Finally, a corresponding numerical algorithm and some illustrated examples are presented.