On matrix inverse eigenvalue problems

This paper discusses the matrix inverse eigenvalue problem: given matrices , find scalars such that

has the prescribed eigenvalues . It is proven that this problem is equivalent to the joint eigenpair problem of a family of related matrix pencils. Furthermore, the conditions under which this problem possesses a solution are given. A method for solving the joint eigenpair problem of a family of matrix pencils is then presented, which, in turn, gives all solutions of the problem. Some results in this paper may be of individual interest.