Inverse problems for (R,S)(R,S)-symmetric matrices in structural dynamic model updating

Let R,S∈Cn×n be nontrivial involutions, i.e., R=R−1≠±InR=R−1≠±In and S=S−1≠±InS=S−1≠±In. A matrix A∈Cn×nA∈Cn×n is referred to as (R,S)(R,S)-symmetric if and only if RAS=ARAS=A. The set of all (R,S)(R,S)-symmetric matrices of order nn is denoted by Csn×n(R,S). Given a full column rank matrix X∈Cn×mX∈Cn×m, a matrix B∈Cm×mB∈Cm×m and a matrix A∗∈Cn×nA∗∈Cn×n. In structural dynamic model updating, we usually consider the sets S1={A∣A∈Csn×n(R,S),XHAX=B} and S2={A∣A∈Csn×n(R,S),‖XHAX−B‖=min} in the Frobenius norm sense, where the superscript HH denotes conjugate transpose. Then we characterize the unique matrices A˜=argminA∈S1‖A−A∗‖ and Â=argminA∈S2‖A−A∗‖ when R=RHR=RH and S=SHS=SH. By using the generalized singular value decomposition (GSVD), the necessary and sufficient conditions for the non-emptiness of S1S1 and the general representations of the elements in S1S1 and A˜ are derived, respectively. The analytical expressions of A∈S2A∈S2 and  are also obtained by using the GSVD, the canonical correlation decomposition (CCD) and the projection theorem. Finally, a corresponding numerical algorithm and some illustrated examples are presented.