Solution of the Inverse Eigenvalue Problem for Certain (Anti-) Hermitian Matrices Using Newton's Method

摘要

Using distinct non zero diagonal elements of a Hermitian (or anti Hermitian) matrix as independent variables of the characteristicpolynomial function, a Newton's algorithm is developed for the solution of the inverse eigenproblem given distinct nonzero eigenvalues. It is found that if a 2$ imes$2 singular Hermitian (or singular anti Hermitian) matrix of rank one is used as the initial matrix, convergence to an exact solution is achieved in only one step. This result can be extended to $n imes n$ ?matrices provided the target eigenvalues are respectively of ?multiplicities $p$ and $q$ with $p+q=n$ and $ 1 leq p,q < n$. Moreover, ?the initial matrix would be of rank one and would have only two distinct corresponding nonzero diagonal elements, the rest being repeated. To illustrate the result, numerical examples are given for the cases $n=2, 3$ and $4$.