Solution of Inverse Eigenvalue Problem of Certain Singular Hermitian Matrices

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We investigate solutions to the Inverse Eigenvalue Problem (IEP) of certain singular Hermitian matrices. Based on a solvability lemma, we propose an algorithm to reconstruct such matrices from their eigenvalues. That is, we develop algorithms and prove that they solve 𝑛×𝑛, singular Hermitian matrices of 𝑟𝑎𝑛𝑘𝑟. In the case of 𝑛×𝑛 matrix, the number of independent matrix elements would reduce to the extent that there would be an isomorphism between the elements and the nonzero eigenvalues. We initiate a differential geometry and numerical analytic interpretation of the Inverse Eigenvalue problem for Hermitian matrices using fibre bundle with structure group 𝑂(𝑛). In particular, Newton type algorithm is developed to construct non singular symmetric matrices using certain singular symmetric matrices as the initial matrices for the iteration.
A Thesis submitted to the Department of Mathematics, Kwame Nkrumah Unversity of Science and Technology in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in Mathematics, August-2012