Bilinear Forms Derived from Lipschitzian Elements in Clifford Algebras

In every Clifford algebra Cl(V, q), there is a Lipschitz monoid (or semi-group) Lip(V, q), which is in most cases the monoid generated by the vectors of V. This monoid is useful for many reasons, not only because of the natural homomorphism from the group GLip(V, q) of its invertible elements onto the group O(V, q) of orthogonal transformations. From every non-zero a is an element of Lip(V, q), we can derive a bilinear form phi on the support S of a in V; it is q-compatible: phi(x, x) = q(x) for all x is an element of S. Conversely, every q-compatible bilinear form on a subspace S of V can be derived from an element a is an element of Lip(V, q) which is unique up to an invertible scalar; and a is invertible if and only if phi is non-degenerate. This article studies the relations between a, phi and (when a is invertible) the orthogonal transformation g derived from a; it provides both theoretical knowledge and algorithms. It provides an effective tool for the factorization of lipschitzian elements, based on this theorem: if (v1, v2, . . . , vs) is a basis of S, then a = kappa v1v2 . . . vs (for some invertible scalar kappa) if and only if the matrix of phi in this basis is lower triangular. This theorem is supported by an algorithm of triangularization of bilinear forms.