Some Inverse Eigenvalue Problems For Structured Matrices And Their Associated Problems On Matrix Equations

Inverse eigenvalue problems concern the reconstruction of matrices with certain spe-cial structures from prescribed complete or partial spectral data.This dissertation shall divide the discussions into five categories:submatrix constrained left and right inverse eigenvalue problem for generalized centrohermitian matrices,the solutions to matrix equa-tions AX = B,YA = D with k-involutory symmetries,Procrustes problems and inverse eigenproblems for multilevel block α-circulants,submatrix constrained least-squares in-verse problem for symmetric matrices,pseudo-Jacobi inverse eigenvalue problems.The specific results covered in each category are in the following:1.An undamped non-gyroscopic model can be discretized as the left and right inverse eigenvalue problem of a certain structured matrix.When the structured matrix is generalized centrohermitian,we discuss the problem with a submatrix constraint.By using Moore-Penrose generalized inverse,singular value decomposition and general-ized singular value decomposition we get necessary and sufficient conditions such that the constrained problem is solvable,and a general representation of the solutions is presented.In addition,an analytical expression of the optimal approximate solution in the Frobenius norm is obtained,and the corresponding numerical algorithm is designed to compute the solution.2.Let R and S be two nontrivial k-involutions.We discuss the properties of(R,S,μ)-symmetric and(R,S,α,μ)-symmetric matrices and denote G as the set of all these matrices.We characterize the least-squares solutions A∈F of ‖AX-B‖2 + ‖YA-D‖2 = min(Frobenius norm)under the assumption that R and S are unitary.Among the set of all such least-squares solutions,we find the optimal approximate matrix A that minimizes ‖A-G‖ to a given unstructural matrix G.Moreover,we also present the necessary and sufficient conditions such that AX = B.YA = D is consistent in G and characterize the consistent solution set.Finally,a numerical algorithm is proposed to compute the optimal approximate solution and illustrative numerical examples are given3.Let n =(n1,n2,…,nk)and α-(α1,α2,…,αk)be integer k-tuples.We discuss the properties of multilevel block α-circulants.When gcd(α,n)= 1 and ged(α,n)(?)1,we then respectively study the Procrustes problems,inverse eigenvalue problems and their optimal approximation problem of these matrices.By using the related results,we can design an artificial Hopfield neural network system,possessing the prescribed equilibria,such that its Jacobian matrix has the constrained multilevel block rx-circulative structure.Finally,some examples are employed to illustrate the effectiveness of the proposed results.4.In structural dynamic model updating,it requires us to compute the least-squares approximations of matrix equation XT AX =B so as to correct the measured mass or stiff matrices.We firstly obtain the least-squares symmetric solutions of this equation with a trailing principal submatrix A0 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 according to a given matrix A*with AO as its trailing principal submatrix.Finally,a corresponding numerical algorithm is established and some numerical examples are presented to verify its feasibility5.The spectral theory and inverse eigenvalue problems for Jacobi matrices is extended to the cases of a certain pseudo-Jacobi matrices J(n,r,β)in the non-self-adjoint set-ting.We investigate the reconstruction of a pseudo-Jacobi matrix from its spectrum and the spectra of two complementary principal matrices.Firstly,two complemen-tary principal matrices are both recovered by using Lanczos algorithm.Then an algorithm for reconstructing the specific pseudo-Jacobi matrices is provided and il-lustrative numerical experiments are performed.