An inverse quadratic eigenvalue problem for damped structural systems (Q955932)

Summary: We first give a representation of the general solution of the following inverse quadratic eigenvalue problem (IQEP): given \(\Lambda=\text{diag}\{\lambda_1,\dots,\lambda_p\}\in\mathbb C^{p\times p}\), \(X=[x_1,\dots,x_p]\in\mathbb C^{n\times p}\), and both \(\Lambda\) and \(X\) are closed under complex conjugation in the sense that \(\lambda_{2j}=\lambda_{2j-1}\in\mathbb C\), \(x_{2j}=\overline x_{2j-1}\in \mathbb C^n\) for \(j=1,\dots,l\), and \(\lambda_k\in\mathbb R\), \(x_k\in\mathbb R^n\) for \(k=2l+1,\dots,p\), find real-valued symmetric \((2r+1)\)-diagonal matrices \(M\), \(D\) and \(K\) such that \(MX\Lambda^2+DX\Lambda+KX=0\). We then consider an optimal approximation problem: given real-valued symmetric \((2r+1)\)-diagonal matrices \(M_a,D_a,K_a\in\mathbb R^{n\times n}\), find \((\widehat M,\widehat D,\widehat K)\in\mathbb S_E\) such that \(\|\widehat M-\widehat M_a\|^2+\|\widehat D-D_a\|^2+\|\widehat K-K_a\|^2=\inf_{(M,D,K)\in\mathbb S_E}(\|M-M_a\|^2+\|D-D_a\|^2+\|K-K_a\|^2)\), where \(S_E\) is the solution set of IQEP. We show that the optimal approximation solution \((\widehat M,\widehat D,\widehat K)\) is unique and derive an explicit formula for it.