### Speaker

Sylvie Icart
(I3S Université Côte d’Azur)

### Description

It is well-known that a Hermitian matrix can be diagonalized by means of a
unitary matrix. The aim of this talk is to present the extension of this result
to polynomial matrices, known as PEVD (Polynomial Eigen Value Decomposition)
[1], occurring e.g. in blind equalization.
In this context, the eigenvalues are polynomials instead of scalars. Moreover,
polynomials are Laurent polynomials, this means with positive and negative
exponents. We will show that in this framework, one can still define unimodular
matrices, Smith form and invariant polynomials. We will present the difference
between the order (or length) and the degree of a polynomial matrix.
Extending polynomial paraconjugation to polynomial matrix, one defines
para-hermitianity. We give some properties of these matrices. Then, we will
show that diagonalization of para-hermitian matrices is not always possible.
Furthermore we will present some results found in the literature on how to
approximate a PEVD.
[1] J. G. McWhirter, P. D. Baxter, T. Cooper, S. Redif, and J. Foster, An
EVD algorithm for Para-Hermitian polynomial matrices, IEEE Transactions
on Signal Processing, vol. 55, no. 6, June 2007, pp. 2158-2169.