Left And Right Inverse Problem And Its Related Matrix Equations For Several Class Of Special Matrices

The constrained matrix equation problem is, in a constrained matrix set, finding a solution of the matrix equation. This problem is one of the topics of very active research in the computational mathematics, and has been widely used in structural design, system identification, principal component analysis, exploration and remote sensing, biology, electricity, solid mechanics, molecular spectroscopy, structural dynamics, automatics control theory, vibration theory, circuit theory, and so on. This Ph.D. Thesis mainly studies following problems.1.Left and right inverse eigenpairs problem is a special inverse eigenvalue problem, i.e. giving partial left and right eigenpairs (eigenvalue and corresponding eigenvector), (λ_i,x_i),i = 1,… ,h;(μ_j,y_j),j = 1,… ,l, a special matrix set S ∈ R^n×n, finding A ∈ S such thatLeft and right inverse eigenpairs problem, which mainly arise in perturbation analysis of matrix eigenvalue, and in recursive matters, have profound application background. This Ph.D.Thesis considers the left and right inverse eigenpairs problem of reflexive(or anti- reflexive) matrices, symmetrizable matrices, persymmet-ric(or skew-persymmetric) matrices, generalized Hamiltonian(or generalized skew-Hamiltonian) matrices, and the related optimal approximation problem. By using special properties of eigenvalue and eigenvector of these matrices, rational wording of each problem is given, and the necessary and sufficient conditions for the existence of and the expressions for the problem are derived. The numerical algorithm and examples to solve the problem are also given.2.The constrained linear matrix equations problem, i.e. giving matrix A ∈ C^h×n,B ∈ C^h×m,C ∈ C^m×l,D ∈ C^n×l, a special matrix set 5 ∈ C^n×m, findingmatrix X ∈ S such thatThis paper discusses reflexive(or anti-reflexive) solutions, generalized reflexive(or generalized skew- reflexive) solutions, symmetrizable solutions, persymmetry(or skew-persymmetry) solutions, generalized Hamiltonian( or generalized skew-Hamiltonian) solutions of matrix equations (AX = B, XC = D), and the related optimal approximation problems. With special properties of these matrices, equivalent equationsof matrix equations (AX = 5, XC =- D) is found, and the necessary and sufficient conditions for the existence of and the expressions for the above problems are obtained. In addition,the numerical algorithm and examples to solve the problem are given.3.This Ph.D.Thesis establishes the least-squares reflexive(or skew-reflexive) solutions, the least-squares generalized reflexive(or generalized skew- reflexive) solutions, the least-squares persymmetry(or skew-persymmetry) solutions, the least-squares generalized Hamiltonian( or generalized skew-Hamiltonian) solutions of matrix equations (AX — B,XC = D). By using the special structure, general expression, or special properties of these matrices, the invariance of the Frobenius norm under orthogonal transformations, the general solutions of above problems and the unique approximation solut: on of related optimal approximation problem are obtained. In addition, the numerical algorithm and examples to solve the problem are given.4.This Ph.D.Thesis further studies the left and right inverse eigenpairs problem and matrix equations problem on the linear manifold, presents the necessary and sufficient conditions for the existence of and the expressions for the above problems. The related optimal approximation problem is also studied, and the numerical algorithm and examples to solve the problem are also given.5.The matrix extension problem is, under some constrained conditions, constructing a matrices X with a given matrix X$ as its submatrix. This Ph.D.Thesis respectively studies the real matrix solutions, reflexive solutions, persymmetry solutions of matrix equations (AX = B, XC = D) with submatrix constrained and its optimal approximation problem. By using the singular value decomposition of matrices, the quotient singular valu€ decomposition of matrix pairs, the canonical correlation decomposition of matrix pairs, the necessary and sufficient conditions for the existence of and the expressions for the above problems are derived, and the numerical algorithm and examples to solve the problem are also given.This Ph.D.Thesis is supported by the Natural Science Foundation of China.This Ph.D.Thesis is typeset by software LATAX2e.