Solution of Inverse Eigenvalue Problem of Certain Singular Hermitian Matrices
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Date
2012-12-18
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Abstract
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.
Description
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