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

An n×nn×n matrix AA is said to be MM-symmetric if xT(A−AT)=0xT(A−AT)=0 for all x∈R(M)x∈R(M), where M∈Rn×pM∈Rn×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 ‖XTAX−C‖‖XTAX−C‖ where ‖⋅‖‖⋅‖ is the Frobenius norm, X∈Rn×mX∈Rn×m and C∈Rm×mC∈Rm×m are given, and A∈Rn×nA∈Rn×n is a MM-symmetric matrix to be solved. Our algorithm produces a suitable AA such that XTAX=CXTAX=C within finite iteration steps in the absence of roundoff errors, if such an AA exists. We show that the algorithm is stable any case, and we give results of numerical experiments that support this claim.