radical (degenerate form)

The radical of a bilinear form B is the set of vectors that are orthogonal to absolutely everything: rad(B) = the set of all v with B(v, w) = 0 for every w in V. These are the vectors the form simply cannot detect — pair them with anything and you get zero. The radical is always a subspace, and it measures exactly how far the form is from being able to tell vectors apart.

In coordinates the radical is nothing exotic: it is the null space of the matrix B. A vector v lies in the radical iff B v = 0, so rad(B) = null(B). Consequently the dimension of the radical is n - rank(B). The rank of a form is just the rank of its matrix, and it counts how many 'effective' directions the form actually sees.

A form is called degenerate when its radical is nonzero — there is some hidden direction it ignores — and nondegenerate when the radical is just the zero vector. Degeneracy is precisely the z (the number of zero squares) in the signature: a form is degenerate iff z > 0 iff det(B) = 0. The radical is the home of all that degeneracy.

The radical also explains why we often pass to a quotient. Modding out by the radical, V / rad(B), produces a smaller space on which the form becomes nondegenerate, with no loss of real information. This is the standard way to reduce the study of any form to the nondegenerate case — strip away the directions the form can't see, and what remains is a clean, faithful form.