Abstract
This paper addresses the extension of the factorization of a Hermitian matrix by an eigenvalue decomposition (EVD) to the case of a parahermitian matrix that is analytic at least on an annulus containing the unit circle. Such parahermitian matrices contain polynomials or rational functions in the complex 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 can be represented by a power or Laurent series that is absolutely convergent, at least on the unit circle, permitting a direct realization in the time domain. Based on an analysis of the unit circle, we prove that eigenvalues exist as unique and convergent but likely 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 phase response is selected such that the eigenvectors are Holder continuous with $\alpha >\frac{1}{2}$ on the unit circle. In the case of a discontinuous phase response or if spectral majorisation is enforced for intersecting eigenvalues, an absolutely convergent Laurent series solution for the eigenvectors of a parahermitian EVD does not exist. We provide some examples, comment on the approximation of a parahermitian matrix EVD by Laurent polynomial factors, and compare our findings to the solutions provided by polynomial matrix EVD algorithms.
Highlights
INTRODUCTIONThe book by Gohberg et al [25] considers the related factorisation of a self-adjoint matrix A(x) = AH (x), x ∈ R, which can be applied to a parahermitian matrix R(z) on the unit circle via the reparameterisation z = ejΩ
F OR a multi-channel signal x[n] ∈ CM the instantaneous covariance matrix is R = E x[n]xH [n], where E{·} denotes the expectation operator and xH represents the Hermitian transpose of x
The parahermitian eigenvalue decomposition (EVD) problem is mapped to the unit circle, i.e. |z| = 1, in Section III where we look at the conditions under which a function of frequency admits a time series representation
Summary
The book by Gohberg et al [25] considers the related factorisation of a self-adjoint matrix A(x) = AH (x), x ∈ R, which can be applied to a parahermitian matrix R(z) on the unit circle via the reparameterisation z = ejΩ Decompositions such as the Smith and Smith-MacMillan forms for matrices of polynomials and rational functions, respectively, are proven to exist [1], [25], but do not involve paraunitary and parahermitian factors as required for the polynomial EVD in [4], [6].
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