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

An n × n matrix A is said to be M -symmetric if x T ( A − A T ) = 0 for all x ∈ R ( M ) , where M ∈ R n × 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 ‖ X T A X − C ‖ where ‖ ⋅ ‖ is the Frobenius norm, X ∈ R n × m and C ∈ R m × m are given, and A ∈ R n × n is a M -symmetric matrix to be solved. Our algorithm produces a suitable A such that X T A X = 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.