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This paper is mainly concerned with a model updating problem for damped structural systems, which can be described as the following inverse quadratic eigenvalue problem (IQEP) and an optimal approximate problem (OAP). Problem IQEP: Given matrices Λ = diag { λ 1 , ... , λ p } ∈ C p × p and X = [ x 1 , ... , x p ] ∈ C n × p with p ≤ n , λ i ≠ λ j for i ≠ j , i , j = 1 , ... , p , rank (X) = p and both Λ and X being closed under complex conjugation, find real symmetric matrices M, D and K such that M X Λ 2 + D X Λ + K X = 0. Problem OAP: Given real symmetric matrices M a , D a and K a , find (M ^ , D ^ , K ^) ∈ S E such that ‖ M ^ − M a ‖ N 2 + ‖ D ^ − D a ‖ N 2 + ‖ K ^ − K a ‖ N 2 = min (M , D , K) ∈ S E (‖ M − M a ‖ N 2 + ‖ D − D a ‖ N 2 + ‖ K − K a ‖ N 2) , where S E is the solution set of Problem IQEP and ‖ ⋅ ‖ N is a weighted Frobenius norm. The representation of the general solution to Problem IQEP and the explicit formula for the optimal approximate solution of Problem OAP are given by applying the singular value decomposition. A given numerical example shows that the updated model can be in good agreement with the modal test data. [ABSTRACT FROM AUTHOR]

Let S R Ω n × n be the set of all n × n symmetric matrices on subspace Ω , where Ω = { z ∈ R n | G z = 0 , G ∈ R k × n }. The necessary and sufficient conditions for the matrix equations A X = B and X C = D to have a common solution in S R Ω n × n and also an expression for the general common solution are obtained. Further, the associated optimal approximate problem to a given matrix X ~ ∈ R n × n is discussed and the optimal approximate solution is elucidated. Finally, a numerical experiment is presented to validate the accuracy of our result. [ABSTRACT FROM AUTHOR]

The finite element model errors mainly come from the complex parts of the geometry, boundary conditions and stress state of the structure. Therefore, the problem for updating mass and stiffness matrices can be reduced to an inverse problem for symmetric matrices with submatrix constraints (IP-MUP): Let Λ = d i a g (λ 1 , ... , λ p) ∈ R p × p and Φ = [ ϕ 1 , ... , ϕ p ] ∈ R n × p be the measured eigenvalue and eigenvector matrices with rank(Φ) = p. Find n × n symmetric matrices M and K such that K Φ = M Φ Λ , Φ ⊤ M Φ = I p , s. t. M (r) = M 0 , K (r) = K 0 , where M(r) and K(r) are the r × r leading principal submatrices of M and K, respectively. We then consider an optimal approximation problem (OAP): Given n × n symmetric matrices Ma and Ka. Find (M ˆ , K ˆ) ∈ S E such that ∥ K ˆ − K a ∥ 2 + ∥ M ˆ − M a ∥ 2 = min (M , K) ∈ S E (∥ K − K a ∥ 2 + ∥ M − M a ∥ 2) , where S E is the solution set of Problem IP-MUP. In this paper, the solvability condition for Problem IP-MUP is established, and the expression of the general solution of Problem IP-MUP is derived. Also, we show that the optimal approximation solution (M ˆ , K ˆ) is unique and derive an explicit formula for it. [ABSTRACT FROM AUTHOR]

Necessary and sufficient conditions are obtained for a pair of matrix equations AX = C and XB = D to have common Re-nnd and Re-pd solutions and the explicit representations of the general common Re-nnd and Re-pd solutions are given when the stated conditions hold. [ABSTRACT FROM AUTHOR]