The inverse eigenvalue problem for a Hermitian reflexive matrix and the optimization problem

Abstract

The inverse eigenvalue problem and the associated optimal approximation problem for Hermitian reflexive matrices with respect to a normal {k +1}-potent matrix are considered. First, we study the existence of the solutions of the associated inverse eigenvalue problem and present an explicit form for them. Then, when such a solution exists, an expression for the solution to the corresponding optimal approximation problem is obtained.

Mathematical modeling and computational methods

“…Secondly, the problem is approached directly and an explicit expression for the n-th power of the system matrix is obtained. The inverse eigenvalue problem and the associated optimal approximation problem for Hermitian reflexive matrices with respect to a normal {k + 1}-potent matrix is analyzed in the work of S. Gigola et al [37]. The singular dispersion matrix within Gauss-Helmert model is considered in [38] by F. Neitzel et al In this contribution the emphasis is shifted towards establishing necessary and sufficient conditions for a unique residual vector, along with a unique estimate of type Best Linear Uniformly Minimum Bias Estimate for the parameter vector.…”

The Centrosymmetric Matrices of Constrained Inverse Eigenproblem and Optimal Approximation Problem

“…Centrosymmetric matrices are applied in information theory, linear system theory, and numerical analysis theory [9]. e unconstrained centrosymmetric matrices' problems have been discussed [9][10][11][12][13][14], a class of unconstrained matrices' inverse eigenproblems has been obtained [15][16][17][18], and the constrained inverse eigenproblems have been discussed [19][20][21][22], but only when the eigenvalues are real or imaginary numbers. For general real matrices, the eigenvalues are not necessarily real or imaginary numbers, so when the eigenvalue is complex, it is difficult to find the constraint solution.…”

BCR Algorithm for Solving Quadratic Inverse Eigenvalue Problems for Partially Bisymmetric Matrices

“…For example, the inverse eigenvalue problem of centro‐symmetric matrices with a submatrix constraint initially occur in the design of Hopfield neural networks, civil engineering, and aviation . The literature on the solvability conditions and numerical approximation of solutions of inverse eigenvalue problems is large and still growing rapidly . In , the sufficient and necessary conditions for the inverse eigenvalue problem with Hermitian, generalized skew‐Hamiltonian, and Hermitian‐generalized Hamiltonian matrices were proposed.…”