Generalized inverse problems for part symmetric matrices on a subspace in structural dynamic model updating

Abstract

An $n\times n$ matrix $A$ is said to be $M$-symmetric if $x^T(A-A^T)=0$ for all $x\in \mathcal{R}\left(M\right)$, where $M\in {\mathbb{R}}^{n\times p}$ is given. In this paper, by extending the idea of the conjugate gradient least squares (CGLS) method, we construct an iterative method for solving a generalized inverse eigenvalue problem: minimizing $\Vert X^{T}AX-C\Vert$ where $\Vert \cdot \Vert$ is the Frobenius norm, $X\in {\mathbb{R}}^{n\times m}$ and $C\in {\mathbb{R}}^{m\times m}$ are given, and $A\in {\mathbb{R}}^{n\times n}$ is a $M$-symmetric matrix to be solved. Our algorithm produces a suitable $A$ such that $X^{T}AX=C$ within finite iteration steps in the absence of roundoff errors, if such an $A$ exists. We show that the algorithm is stable any case, and we give results of numerical experiments that support this claim.