Submatrix constrained least-squares inverse problem for symmetric matrices from the design of vibrating structures

Abstract

Given a full column rank matrix X ∈ R n × m , a matrix B ∈ R m × m and a symmetric matrix A 0 ∈ R p × p . In structural dynamic model updating, Yuan and Dai (2007) considered the matrix equation X T A X = B with a leading principal submatrix A 0 constraint. For a given matrix A ∗ ∈ R n × n , they updated the mass and stiffness matrix in the Frobenius norm sense such that the corrected matrices satisfy the generalized eigenvalue equation and orthogonality conditions. But due to measurement errors, the measured mass and stiffness matrices will not always satisfy these requirements. Since they still contain some useful information, we would like to retrieve their least-squares approximations to correct these matrices. Then we obtain the least-squares symmetric solutions of the equation X T A X = B with a trailing principal submatrix A 0 constraint by using the matrix differential calculus and canonical correlation decomposition. Furthermore, by applying the generalized singular value decomposition and projection theorem we get the best Frobenius norm approximate symmetric solution of this equation according to a given matrix A ∗ ∈ R n × n with A 0 as its trailing principal submatrix. Finally, a numerical algorithm for computing the best approximate solution is established. Some illustrated numerical examples are also presented.