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Inverse problems for \((R,S)\)-symmetric matrices in structural dynamic model updating. (English) Zbl 1443.15013

Summary: Let \(R,S\in\mathbb{C}^{n\times n}\) be nontrivial involutions, i.e., \(R=R^{-1}\neq\pm I_n\) and \(S=S^{-1}\neq\pm I_n\). A matrix \(A\in\mathbb{C}^{n\times n}\) is referred to as \((R,S)\)-symmetric if and only if \(RAS=A\). The set of all \((R,S)\)-symmetric matrices of order \(n\) is denoted by \(\mathcal C_s^{n\times n}(R,S)\). Given a full column rank matrix \(X\in\mathbb{C}^{n\times m}\), a matrix \(B\in\mathbb{C}^{m\times m}\) and a matrix \(A^\ast\in\mathbb{C}^{n\times n}\). 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 denotes conjugate transpose. Then we characterize the unique matrices \(\widetilde{A}=\arg\min\limits_{A\in\mathcal S_1}\Vert A-A^\ast\Vert\) and \(\hat{A}=\arg\min\limits_{A\in\mathcal S_2}\Vert A-A^\ast\Vert\) when \(R=R^{\text{H}}\) and \(S=S^{\text{H}}\). By using the generalized singular value decomposition (GSVD), the necessary and sufficient conditions for the non-emptiness of \(\mathcal S_1\) and the general representations of the elements in \(\mathcal S_1\) and \(\widetilde{A}\) are derived, respectively. The analytical expressions of \(A\in\mathcal S_2\) and \(\hat{A}\) 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.

MSC:

15A29 Inverse problems in linear algebra
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