We use the language of signal flow graph representation of digital filter structures to solve three purely mathematical problems, including fast inversion of certain polynomial–Vandermonde matrices, deriving an analogue of the Horner and Clenshaw rules for polynomial evaluation in a (H, m)–quasiseparable basis, and computation of eigenvectors of (H, m)– quasiseparable classes of matrices. While...

In this paper, we present a new and efficient algorithm for computing minimal polynomial of matrices over algebraic extension fields using the Gröbner bases technique. We have implemented our algorithm in Maple and we evaluate its performance and compare it to the performance of the function MinimalPolynomial of Maple 15 and also of the Bia las algorithm as a new algorithm to compute minimal po...

Let F be an infinite field and let f a nonzero multilinear polynomial with coefficients in . We prove that for every positive integer d there exists s such ( M ) , the image of contains all trace zero × matrices. In particular, algebra finitary matrices

Connection matrices for graph parameters with values in a field have been introduced by M. Freedman, L. Lovász and A. Schrijver (2007). Graph parameters with connection matrices of finite rank can be computed in polynomial time on graph classes of bounded tree-width. We introduce join matrices, a generalization of connection matrices, and allow graph parameters to take values in the tropical ri...

We propose two simple upper bounds for the joint spectral radius of sets of nonnegative matrices. These bounds, the joint column radius and the joint row radius, can be computed in polynomial time as solutions of convex optimization problems. We show that for general matrices these bounds are within a factor 1/n of the exact value, where n is the size of the matrices. Moreover, for sets of matr...

We give modular algorithms to compute row-reduced forms, weak Popov forms, and Popov forms of polynomial matrices, as well as the corresponding unimodular transformation matrices. Our algorithms improve on existing fraction-free algorithms. In each case we define lucky homomorphisms, determine the appropriate normalization, as well as bound the number of homomorphic images required. The algorit...

We give a probabilistic interpretation for the Barnes G-function which appears in random matrix theory and in analytic number theory in the important moments conjecture due to Keating-Snaith for the Riemann zeta function, via the analogy with the characteristic polynomial of random unitary matrices. We show that the Mellin transform of the characteristic polynomial of random unitary matrices an...

Computational Algebra methods have been used successfully in various problems in many fields of Mathematics. Computational Algebra encompasses a set of powerful algorithms for studying ideals in polynomial rings and solving systems of nonlinear polynomial equations efficiently. The theory of Gröbner bases is a cornerstone of Computational Algebra, since it provides us with a constructive way of...

We define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions are given by the semi-circle law and the Marčenko-Pastur law are special cases. Algebraicity of a ...