Many standard conversion matrices between coefficients in classical orthogonal polynomial expansions can be decomposed using diagonally-scaled Hadamard products involving Toeplitz and Hankel matrices. This allows us to derive O(N(logN)) algorithms, based on the fast Fourier transform, for converting coefficients of a degree N polynomial in one polynomial basis to coefficients in another. Numeri...

This paper presents several new results on the inversion of full normal rank nonsquare polynomial matrices. New analytical right/left inverses of polynomial matrices are introduced, including the so-called τ -inverses, σ-inverses and, in particular, S-inverses, the latter providing the most general tool for the design of various polynomial matrix inverses. The applicationoriented problem of sel...

In the context of multivariate signal processing, factorizations involving so-called para-unitary matrices are relevant as well demonstrated in the book of Vaidyanathan [11], or [4, 1] and more recently in a series of papers by McWhirter and co-authors [5, 6]. However, known factorizations of matrix polynomials, such as the Smith form [10], involve unimodular matrices but usual factorizations s...

Let k be a field. Then Gaussian elimination over k and the Euclidean division algorithm for the univariate polynomial ring k[x] allow us to write any matrix in SLn(k) or SLn(k[x]), n ≥ 2, as a product of elementary matrices. Suslin’s stability theorem states that the same is true for the multivariate polynomial ring SLn(k[x1, . . . , xm]) with n ≥ 3. As Gaussian elimination gives us an algorith...

We present a concise semidefinite formulation for the problem of minimizing a polynomial over a semi-algebraic set defined by polynomial equalities and inequalities. When the polynomial equalities define a radical ideal I with a finite variety, this representation involves combinatorial moment matrices, indexed by a basis of R[x1, . . . , xn]/I. The arguments are elementary and extend known fac...

The Amitsur-Levitzki theorem asserts that Mn(F ) satisfies a polynomial identity of degree 2n. (Here, F is a field and Mn(F ) is the algebra of n × n matrices over F ). It is easy to give examples of subalgebras of Mn(F ) that do satisfy an identity of lower degree and subalgebras of Mn(F ) that satisfy no polynomial identity of degree ≤ 2n − 2. In this paper we prove that the subalgebras of n ...

Boolean circuits of polynomial size and poly-logarithmic depth are given for computing the Hermite and Smith normal forms of polynomial matrices over finite fields and the field of rational numbers. The circuits for the Smith normal form computation are probabilistic ones and also determine very efficient sequential algorithms. Furthermore, we give a polynomial-time deterministic sequential alg...

In this paper, we present an algorithm for computing the characteristic polynomial of the pencil (A ? sE). It is shown that after a preliminary reduction of the matrices A and E to, respectively, an upper Hessenberg and an upper triangular matrix, the problem of computing the characteristic polynomial is transformed to the solution of certain triangular systems of linear algebraic equations. We...

In this paper we prove that a vertex-centered automorphism of a tree gives a proper factor of the characteristic polynomial of its distance or adjacency matrix. We also show that the characteristic polynomial of the distance matrix of any graph always has a factor of degree equal to the number of vertex orbits of the graph. These results are applied to full k-ary trees and some other problems. ...

We show that error and erasure decoding of `-Interleaved Gabidulin codes, as well as list-` decoding of Mahdavifar–Vardy codes can be solved by row reducing skew polynomial matrices. Inspired by row reduction of F[x] matrices, we develop a general and flexible approach of transforming matrices over skew polynomial rings into certain normal forms. We apply this to solve generalised shift registe...