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

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.