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 $R,S\in {\mathbb{C}}^{n\times n}$ be nontrivial involutions, i.e., R=R⁻¹≠±I_n$R=R^{-1}\ne \pm I_n$ and S=S⁻¹≠±I_n$S=S^{-1}\ne \pm I_n$. A matrix A∈C^n×n$A\in {\mathbb{C}}^{n\times n}$ is referred to as (R,S)$(R,S)$-symmetric if and only if RAS=A$RAS=A$. The set of all (R,S)$(R,S)$-symmetric matrices of order n$n$ is denoted by ${\mathcal{C}}_s^{n\times n}(R,S)$. Given a full column rank matrix X∈C^n×m$X\in {\mathbb{C}}^{n\times m}$, a matrix B∈C^m×m$B\in {\mathbb{C}}^{m\times m}$ and a matrix A^∗∈C^n×n$A^{\ast }\in {\mathbb{C}}^{n\times n}$. In structural dynamic model updating, we usually consider the sets ${\mathcal{S}}_1=\{A\mid A\in {\mathcal{C}}_s^{n\times n}(R,S),X^{\text{H}}AX=B\}$ and ${\mathcal{S}}_2=\{A\mid A\in {\mathcal{C}}_s^{n\times n}(R,S),\Vert X^{\text{H}}AX-B\Vert =min\}$ in the Frobenius norm sense, where the superscript H$\text{H}$ denotes conjugate transpose. Then we characterize the unique matrices $\tilde{A}=arg\underset{A\in {\mathcal{S}}_1}{min}\Vert A-A^{\ast }\Vert$ and $\widehat{A}=arg\underset{A\in {\mathcal{S}}_2}{min}\Vert A-A^{\ast }\Vert$ when R=R^H$R=R^{\text{H}}$ and S=S^H$S=S^{\text{H}}$. By using the generalized singular value decomposition (GSVD), the necessary and sufficient conditions for the non-emptiness of S₁${\mathcal{S}}_1$ and the general representations of the elements in S₁${\mathcal{S}}_1$ and $\tilde{A}$ are derived, respectively. The analytical expressions of A∈S₂$A\in {\mathcal{S}}_2$ and $\widehat{A}$ 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.