Submatrix Constrained Inverse Eigenvalue Problem involving Generalised Centrohermitian Matrices in Vibrating Structural Model Correction

Generalised centrohermitian and skew-centrohermitian matrices arise in a variety of applications in different fields. Based on the vibrating structure equation M(x) over dot + (D+G)(x) over dot + Kx = f(t) where M, D, G, K are given matrices with appropriate sizes and x is a column vector, we design a new vibrating structure mode. This mode can be discretised as the left and right inverse eigenvalue problem of a certain structured matrix. When the structured matrix is generalised centrohermitian, we discuss its left and right inverse eigenvalue problem with a submatrix constraint, and then get necessary and sufficient conditions such that the problem is solvable. A general representation of the solutions is presented, and an analytical expression for the solution of the optimal approximation problem in the Frobenius norm is obtained. Finally, the corresponding algorithm to compute the unique optimal approximate solution is presented, and we provide an illustrative numerical example.