The finite element model errors mainly come from the complex parts of the geometry, boundary conditions and stress state of the structure. Therefore, the problem for updating mass and stiffness matrices can be reduced to an inverse problem for symmetric matrices with submatrix constraints (IP-MUP): Let $\mathrm{\Lambda }=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}({\lambda }_1,\dots ,{\lambda }_p)\in {\mathbb{R}}^{p\times p}$ and $\mathrm{\Phi }=[{\varphi }_1,\dots ,{\varphi }_p]\in {\mathbb{R}}^{n\times p}$ be the measured eigenvalue and eigenvector matrices with $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\mathrm{\Phi }\right)=p$. Find $n\times n$ symmetric matrices M and K such that $K\mathrm{\Phi }=M\mathrm{\Phi }\mathrm{\Lambda },{\mathrm{\Phi }}^{\mathrm{\top }}M\mathrm{\Phi }=I_p,\mathrm{s}.\mathrm{t}.M_{\left(r\right)}=M_0,K_{\left(r\right)}=K_0,$ where $M_{\left(r\right)}$ and $K_{\left(r\right)}$ are the $r\times r$ leading principal submatrices of M and K, respectively. We then consider an optimal approximation problem (OAP): Given $n\times n$ symmetric matrices $M_a$ and $K_a$. Find $(\hat{M},\hat{K})\in {\mathcal{S}}_E$ such that $\parallel \hat{K}-K_a{\parallel }^2+\parallel \hat{M}-M_a{\parallel }^2=\underset{(M,K)\in {\mathcal{S}}_E}{min}(\parallel K-K_a{\parallel }^2+\parallel M-M_a{\parallel }^2)$, where ${\mathcal{S}}_E$ is the solution set of Problem IP-MUP. In this paper, the solvability condition for Problem IP-MUP is established, and the expression of the general solution of Problem IP-MUP is derived. Also, we show that the optimal approximation solution $(\hat{M},\hat{K})$ is unique and derive an explicit formula for it.

有限元模型误差主要来自几何结构的复杂部分，边界条件和结构的应力状态。因此，对于具有子矩阵约束的对称矩阵（IP-MUP），可以将更新质量和刚度矩阵的问题简化为一个反问题：$\mathrm{\Lambda }=\mathrm{d}\mathrm{一世}\mathrm{一种}\mathrm{G}（{\lambda }_{\mathrm{1个}}，\dots ，{\lambda }_p）\in {\mathbb{[R}}^{p\times p}$ 和 $\mathrm{\Phi }=[{\varphi }_{\mathrm{1个}}，\dots ，{\varphi }_p]\in {\mathbb{[R}}^{ñ\times p}$ 是测量的特征值和特征向量矩阵 $\mathrm{[R}\mathrm{一种}\mathrm{ñ}\mathrm{ķ}（\mathrm{\Phi }）=p$。找$ñ\times ñ$对称矩阵中号和ķ使得$ķ\mathrm{\Phi }=\mathrm{中号}\mathrm{\Phi }\mathrm{\Lambda }，{\mathrm{\Phi }}^{\mathrm{\top }}\mathrm{中号}\mathrm{\Phi }={\mathrm{一世}}_p，\mathrm{s}。\mathrm{Ť}。{\mathrm{中号}}_{（\mathrm{[R}）}={\mathrm{中号}}_0，{ķ}_{（\mathrm{[R}）}={ķ}_0，$ 哪里 ${\mathrm{中号}}_{（\mathrm{[R}）}$ 和 ${ķ}_{（\mathrm{[R}）}$ 是 $\mathrm{[R}\times \mathrm{[R}$M和K的前导主要子矩阵。然后，我们考虑最佳逼近问题（OAP）：$ñ\times ñ$ 对称矩阵 ${\mathrm{中号}}_{\mathrm{一种}}$ 和 ${ķ}_{\mathrm{一种}}$。找$（\widehat{\mathrm{中号}}，\widehat{ķ}）\in {\mathcal{小号}}_{Ë}$ 这样 $\parallel \widehat{ķ}-{ķ}_{\mathrm{一种}}{\parallel }^2+\parallel \widehat{\mathrm{中号}}-{\mathrm{中号}}_{\mathrm{一种}}{\parallel }^2=\underset{（\mathrm{中号}，ķ）\in {\mathcal{小号}}_{Ë}}{分}（\parallel ķ-{ķ}_{\mathrm{一种}}{\parallel }^2+\parallel \mathrm{中号}-{\mathrm{中号}}_{\mathrm{一种}}{\parallel }^2）$，在哪里 ${\mathcal{小号}}_{Ë}$是问题IP-MUP的解决方案集。本文建立了问题IP-MUP的可解性条件，并推导了问题IP-MUP的一般解的表示。此外，我们证明了最佳逼近解$（\widehat{\mathrm{中号}}，\widehat{ķ}）$ 是唯一的，并为此导出一个明确的公式。