Given an isometry f of a quadratic, symplectic or unitary space V with bilinear or sesquilinear form β, the Wall form of f is the form θ defined on the image I of 1 - f by θ(u, v) = β(w, v), where u = w(1 - f). In general, the Wall form is not reflexive.
The space of the Wall form of the isometry f and its embedding in V.
The inverse of WallForm. This is an isometry corresponding to the embedding μ : I to V, where V is a quadratic, symplectic or unitary space.
An isometry f of a quadratic or symplectic space V is Wall-regularif the restriction of 1 - f to the image of 1 - f is invertible. If f is any isometry of V this function returns a Wall-regular element fr and a unipotent element fu such that f = frfu = fu fr.
If V is a vector space with a bilinear form β, a basis e1, e2, ..., en for V is semi-orthogonal if β(ei, ej) = 0 for i < j. This function returns a semi-orthogonal basis with respect to the non-degenerate, non-alternating form attached to V. If the base field is GF(2), the form should be symmetric.
This function returns the space of the generalised Wall form of the similarity f and its embedding in the quadratic space V. Suppose that the quadratic form Q of V is nondegenerate and let β be its polar form. Then Q(vf) = η Q(v) for some η. Suppose that η = ζ2 and let V(f, ζ) denote the ζ-eigenspace of f. The generalised Wall form θ of f is defined on the orthogonal complement of V(f, ζ) by θ(u, v) = β(w, v), where u = ζ w - wf.