Unimodular Form - Reflexivity and Orthogonality

Reflexivity and Orthogonality

Definition: A bilinear form B : V × V → F is called reflexive if B(v, w) = 0 implies B(w, v) = 0 for all v, w in V.

Definition: Let B : V × V → F be a reflexive bilinear form. v, w in V are orthogonal with respect to B if and only if B(v, w) = 0 or B(w, v) = 0.

A form B is reflexive if and only if it is either symmetric or alternating. In the absence of reflexivity we have to distinguish left and right orthogonality. In a reflexive space the left and right radicals agree and are termed the kernel or the radical of the bilinear form: the subspace of all vectors orthogonal with every other vector. A vector v, with matrix representation x, is in the radical of a bilinear form with matrix representation A, if and only if Ax = 0 ↔ xTA = 0. The radical is always a subspace of V. It is trivial if and only if the matrix A is nonsingular, and thus if and only if the bilinear form is nondegenerate.

Suppose W is a subspace. Define the orthogonal complement

For a non-degenerate form on a finite dimensional space, the map W ↔ W⊥ is bijective, and the dimension of W⊥ is dim(V) − dim(W).