rank of a form

Some directions in a space are invisible to a form — feeding them in always returns zero no matter the partner. The rank measures how many directions actually ‘count’: it is the size of the non-trivial part of the form once the blind directions (the radical) are quotiented away. A full-rank form sees everything; a deficient rank means the form is degenerate along some subspace.

Precisely, the rank of a bilinear or quadratic form is the rank of any of its Gram matrices. Because changing basis replaces G by the congruent matrix P^T G P with P invertible, the rank is unchanged — it is a congruence invariant, hence an invariant of the form itself. Equivalently the rank is the dimension of V minus the dimension of the radical, the subspace of vectors orthogonal to all of V.

After diagonalization the rank is simply the number of nonzero diagonal entries. The form is nondegenerate exactly when the rank equals dim V, equivalently when the Gram matrix is invertible, equivalently when the radical is zero. Over R the rank equals p + q where (p, q) is the signature; the difference p - q refines the rank with sign information that the rank alone discards.