Symmetric Bilinear Form - Orthogonality and Singularity

Orthogonality and Singularity

A symmetric bilinear form is always reflexive. Two vectors v and w are defined to be orthogonal with respect to the bilinear form B if, which is, due to reflexivity, equivalent with

The radical of a bilinear form B is the set of vectors orthogonal with every other vector in V. One easily checks that this is a subspace of V. When working with a matrix representation A with respect to a certain basis, v, represented by x, is in the radical if and only if

The matrix A is singular if and only if the radical is nontrivial.

If W is a subset of V, then the orthogonal complement is the set of all vectors orthogonal with every vector in W: it is a subspace of V. When B is non-degenerate, so that the radical of B is trivial, the dimension of = dim(V) − dim(W).