In this thesis, we focus on a particular type of matrices: Laurent polynomial matrices whose elements are Laurent polynomials, ie polynomials with positive and negative powers of $ z $. Such polynomials cannot be associated to a causal filter but they occur when considering the spectrum of finite impulse response filter output. First, we present the properties of Laurent polynomials and Laurent polynomial matrices. We define the L-Smith form which is an extension of the classical Smith form of a polynomial matrix and give a precise definition of the degree and the order of these matrices (sometimes confused notions in the literature). We then study para-Hermitian matrices and para-unitary matrices which are respectively equal to their para-conjugated or whose inverse is equal to the para-conjugated. We develop their properties in terms of particular degree and factorization. In system theory and signal processing, many factorizations of matrices with constant coefficients are involved: QR factorizations (using an orthogonal and a triangular matrix), LU (using two triangular matrices: one lower and one upper), SVD (singular value decomposition using two unitary matrices) EVD (eigenvalue decompositions). In particular, the spectral theorem shows that every hermitian matrix can be diagonalized using a unitary matrix. The Cholesky factorization of a hermitian positive definite matrix uses a triangular matrix and its conjugate transpose. These factorizations cannot be extended to polynomial matrices because the coefficients of these matrices do not belong to a field but a ring (that of Laurent polynomials). We show that in the general case, a polynomial EVD decomposition of a para-Hermitian matrix which is positive definite on the unit circle does not exist, but one can almost-diagonalize these matrices using para-unitary matrices which are continuous on the unit circle. Finally, we show what role para-unitary matrices factorizations plays in blind equalization of convolutive multivariable systems.
Matrices polynomiales et égalisation de canal