In this paper, we will deal with the following two classes of best approximation problems about the linear manifolds: Problem 1. Given matrices $A_1,B_1,C_1,$ and $D_1\in {\mathbb{R}}^{m\times n}$, find $d(L_1,L_2)=\underset{X\in L_1,Y\in L_2}{min}\Vert X-Y\Vert ,$ and find $\hat{X}\in L_1,\hat{Y}\in L_2$ such that $\Vert \hat{X}-\hat{Y}\Vert =d(L_1,L_2)$, where $L_1=\left\{X\in {\mathbb{S}\mathbb{R}}^{n\times n}|\Vert A_{1}X-B_1\Vert =min\right\}$ and $L_2=\left\{Y\in {\mathbb{S}\mathbb{R}}^{n\times n}|\Vert C_{1}Y-D_1\Vert =min\right\}$. Problem 2. Given matrices $A_2,B_2,E_2,F_2\in {\mathbb{R}}^{m\times n}$ and $C_2,D_2,G_2,H_2\in {\mathbb{R}}^{n\times p}$, find $d(L_3,L_4)=\underset{X\in L_3,Y\in L_4}{min}\Vert X-Y\Vert ,$ and find $\tilde{X}\in L_3,\tilde{Y}\in L_4$ such that $\Vert \tilde{X}-\tilde{Y}\Vert =d(L_3,L_4)$, where $L_3=\left\{X\in {\mathbb{R}}^{n\times n}|\Vert A_{2}X-B_2{\Vert }^2+||X{C}_2-D_2{\Vert }^2=min\right\}$ and $L_4=\left\{Y\in {\mathbb{R}}^{n\times n}|\Vert E_{2}Y-F_2{\Vert }^2+||Y{G}_2-H_2{\Vert }^2=min\right\}$. We obtain explicit formulas for $d(L_1,L_2)$ and $d(L_3,L_4),$ and all the matrices in question by using the singular value decompositions and the canonical correlation decompositions of matrices.

In this paper, the necessary and sufficient conditions under which the matrix inequality $C^{\ast }XC\ge D(>D)$ subject to the linear constraint $A^{\ast }XA=B$ is solvable are deduced by means of the spectral decompositions of some matrices and the generalized singular value decomposition of a matrix pair. An explicit expression of the general Hermitian solution is also provided. One numerical example demonstrates the effectiveness of the proposed method.

A direct method for simultaneously updating mass and stiffness matrices of the undamped piezoelectric smart structural models based on incomplete modal measured data is presented. By applying the generalized singular value decomposition and some matrix derivatives, the optimal approximate mass and stiffness matrices which satisfy the required eigenvalue equation and the orthogonality relation are found under the Frobenius norm sense. The method is computationally efficient as neither iteration nor eigenanalysis is required. Numerical results are included to illustrate the effectiveness of the proposed method.

The solvability conditions for the dual matrix equation $AXB=D$ and a pair of dual matrix equations $AX=C$ and $XB=D$ are deduced by applying the singular value decomposition, and the expressions of the general solutions to these dual matrix equations are provided. Furthermore, the minimum-norm solutions of these dual matrix equations are provided. Finally, two numerical experiments are given to validate the accuracy of the results obtained.