No abstract

We present an algorithmic proof of the Cartan-Dieudonné theorem on generalized real scalar product spaces with arbitrary signature. We use Clifford algebras to compute the factorization of a given orthogonal transformation as a product of reflections with respect to hyperplanes. The relationship with the Cartan-Dieudonné-Scherk theorem is also discussed in relation to the minimum number of reflections required to decompose a given orthogonal transformation.

We present a new way to deal with doubly in nite lower Hessenberg matrices based on the representation of the matrices as the sum of their diagonal submatrices. We show that such representation is a simple and useful tool for computation purposes and also to obtain general properties of the matrices related with inversion, similarity, commutativity, and Pincherle derivatives. The diagonal representation allows us to consider the ring of doubly in nite lower Hessenberg matrices over a ring R as a ring of Laurent series in one indeterminate, with coe cients in the ring of R-valued sequences that don't commute with the indeterminate.

No abstract

We obtain some properties of a class of q-hypergeometric orthogonal polynomials with q = −1, described by a uniform parametrization of the recurrence coefficients. We show that our class contains the Bannai-Ito polynomials and other known -1 polynomials. We introduce some new examples of -1 polynomials and also obtain matrix realizations of the Bannai-Ito algebra.