The solvability conditions for the inverse eigenproblems of symmetric and generalized centro-symmetric matrices and their approximations

Let ∥ · ∥ be the Frobenius norm of matrices. We consider (I) the set SE of symmetric and generalized centro-symmetric real n × n matrices Rn with some given eigenpairs (λj, qj) (j = 1, 2, … , m) and (II) the element in SE which minimizes for a given real matrix R∗. Necessary and sufficient conditions for SE to be nonempty are presented. A general form of elements in SE is given and an explicit expression of the minimizer is derived. Finally, a numerical example is reported.