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

Francis T. Oduro

Abstract


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$\times$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 $n \times n$  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 $4$.

Full Text: PDF DOI: 10.5539/jmr.v6n2p64

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.

Journal of Mathematics Research   ISSN 1916-9795 (Print)   ISSN 1916-9809 (Online)

Copyright © Canadian Center of Science and Education

To make sure that you can receive messages from us, please add the 'ccsenet.org' domain to your e-mail 'safe list'. If you do not receive e-mail in your 'inbox', check your 'bulk mail' or 'junk mail' folders.