In this paper, we develop a novel and effective Euclidean algorithm for Laurent polynomial matrix extension (LPME), which is the key of the construction of perfect reconstruction filter banks (PRFBs). The algorithm simplifies the dual-chain approach to the construction of PRFBs in the paper [5]. The algorithm in this paper can also be used in the applications where LPME plays a role.

An efficient algorithm for evaluating the matrix polynomial I + A + A + ••• +A~ is developed. The proposed scheme is simple and eliminates the difficulties encountered in applying a recently reported procedure [11.

Two main approaches are used, nowadays, to compute the roots of a zero-dimensional polynomial system. The rst one involves Grrbner basis computation, and applies to any zero-dimensional system. But, it is performed with exact arithmetic and, usually, big numbers appear during the computation. The other approach is based on resultant formulations and can be performed with oating point arithmetic...

We present the asymptotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank, nullspace, determinant, generic inverse, reduced form [8, 9, 16, 17]. We show that they essentially can be reduced to two computer algebra techniques, minimal basis computations and matrix fraction expansion/reconstruction, and to polynomial matrix multiplication. Such reducti...

For parahermitian polynomial matrices, which can be used, for example, to characterise space-time covariance in broadband array processing, the conventional eigenvalue decomposition (EVD) can be generalised to a polynomial matrix EVD (PEVD). In this paper, a new iterative PEVD algorithm based on sequential matrix diagonalisation (SMD) is introduced. At every step the SMD algorithm shifts the do...

Hilbert's 17th problem concerns expressing polynomials on R as a sum of squares. It is well known that many positive polynomials are not sums of squares; see [R00] [deA preprt] for excellent surveys. In this paper we consider symmetric non-commutative polynomials and call one \matrix positive", if whenever matrices of any size are substituted for the variables in the polynomial the matrix value...

A standard way of dealing with matrix polynomial eigenvalue problems is to use linearizations. Byers, Mehrmann and Xu have recently defined and studied linearizations of dimensions smaller than the classical ones. In this paper, lower bounds are provided for the dimensions of linearizations and strong linearizations of a given m×n matrix polynomial, and particular linearizations are constructed...

Some block Toeplitz methods applied to polynomial matrices are reviewed. We focus on the computation of the structure (rank, null-space, infinite and finite structures) of an arbitrary rectangular polynomial matrix. We also introduce some applications of this structural information in control theory. All the methods outlined here are based on the computation of the null-spaces of suitable block...

We investigate the use of polynomial matrices to give efficient presentations of nonnegative matrices exhibiting prescribed spectral and algebraic behavior.