Summary: Finite element model updating is a procedure to minimize the differences between analytical and experimental results and can be mathematically reduced to solving the following problem. Problem P: Let \(M_a\in\mathbf{SR}^{n\times n}\) and \(K_a\in\mathbf{SR}^{n\times n}\) be the analytical mass and stiffness matrices and \(\Lambda=\mathrm{diag}\{\lambda_1,\dots,\lambda_p\}\in\mathbf R^{p\times p}\) and \(X=[x_1,\dots,x_p]\in\mathbf R^{n\times p}\) be the measured eigenvalue and eigenvector matrices, respectively. Find \((\hat M,\hat K)\in\mathcal S_{MK}\) such that \(\|\hat M-M_a\|^2+\|\hat K-K_a\|^2=\min_{(M,K)\in\mathcal S_{MK}}\) \(\| M-M_a\|^2+\| K-K_a\|^2)\), where \(\mathcal S_{MK}=\{(M,K)\mid X^TMX=I_p\), \(MX\varLambda=KX\}\) and \(\|\cdot\|\) is the Frobenius norm. This paper presents an iterative method to solve Problem P. By the method, the optimal approximation solution \((\hat M,\hat K)\) of Problem P can be obtained within finite iteration steps in the absence of roundoff errors by choosing a special kind of initial matrix pair. A numerical example shows that the introduced iterative algorithm is quite efficient.