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.