On the Existence and Uniqueness of the Eigenvalue Decomposition of a Parahermitian Matrix - 2018


This Project addresses the extension of the factorization of a Hermitian matrix by an eigenvalue decomposition (EVD) to the case of a parahermitian matrix that's analytic a minimum of on an annulus containing the unit circle. Such parahermitian matrices contain polynomials or rational functions in the complicated variable z and arise, e.g., as cross spectral density matrices in broadband array problems. Specifically, conditions for the existence and uniqueness of eigenvalues and eigenvectors of a parahermitian matrix EVD are given, such that these will be represented by a power or Laurent series that is completely convergent, at least on the unit circle, allowing a right away realization within the time domain. Primarily based on an analysis of the unit circle, we have a tendency to prove that eigenvalues exist as unique and convergent but seemingly infinite-length Laurent series. The eigenvectors can have an arbitrary phase response and are shown to exist as convergent Laurent series if eigenvalues are selected as analytic functions on the unit circle, and if the part response is selected such that the eigenvectors are Hölder continuous with a > 1/2 on the unit circle. Within the case of a discontinuous section response or if spectral majorisation is enforced for intersecting eigenvalues, an absolutely convergent Laurent series resolution for the eigenvectors of a parahermitian EVD will not exist. We have a tendency to provide some examples, investigate the approximation of a parahermitian matrix EVD by Laurent polynomial factors, and compare our findings to the solutions provided by polynomial matrix EVD algorithms.

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