On Some Inverse Eigenvalue Problems For Structured Matrices And Matrix Equation Problems

Inverse eigenvalue problems, one of the most important disciplines in the fieldsof numerical algebra, arise in a remarkable variety of applications including mathe-matical physics, particle physics, quantum mechanics, geophysics, molecular spec-troscopy, physical design, parametric recognition, automatic control, and so on.Matrix equation, which is another important subject in numerical algebra,associated with the linear and nonlinear cases, has been widely applied to a settingof varied fields such as biology, optics, electronics, dynamic programming, statistics,system control, etc.In this paper, the following inverse eigenvalue problems and matrix equationproblems are explored.1. A class of inverse eigenvalue problems for sub-periodic Jacobi matrices arediscussed.Problem I Given two real setsλ= {λ₁,λ₂,···,λ_n},μ= {μ₁,μ₂,···,μ_n}, satis-fyingλ₁ <μ₁ <λ₂ <···<λ_n-1 <μ_n-1 <λ_n,and a positive numberβ, find a sub-periodic Jacobi matrix Sn, such thatProblem II Given two real setsλ= {λ₁,λ₂,···,λ_n},μ= {μ₁,μ₂,···,μ_n}, satis-fyingλ₁ <μ₁ <λ₂ <···<λ_n-1 <μ_n-1 <λ_n,and a positive numberβ, find a sub-periodic Jacobi matrix S_n, such thatThe necessary and sufficient conditions for the solvability and uniqueness ofthe problems are obtained, together with the feasible stable numerical algorithms. 2. Several classes of linear matrix equations and the least square problems arediscussed.Problem III Considering the following matrix equationwhere A,B∈R^m×n, ||·||_F means Frobenius norm, .Problem IV Considering the following matrix equationminwhere A∈R^m×n, B∈R^n×r, C∈R^m×r,·F means Frobenius norm, .Problem V Considering the following matrix equationwhere A,B,C∈R^n×n, ||·||_F means Frobenius norm, .A conclusion that the linear matrix equation problems above are solvable whenP are subsets of general Toeplitz matrices, triangular Toeplitz matrices, and sym-metric Toeplitz matrices respectively are gained. Also, the necessary and su?cientconditions for the uniqueness are given. Numerical algorithms and examples aredesired to direct computing procedure and to verify the feasibility.3. A class of nonlinear matrix equations for the matrix square roots are dis-cussedX² = A,where A is an upper triangular toeplitz matrix. The necessary and su?cient con-ditions for the solvability, along with the form and the number of the solutions, areobtained. The conclusion partly improves and extends the related existing results.