Tensor product of quadratic forms

From Wikipedia, the free encyclopedia
Jump to: navigation, search

The tensor product of quadratic forms is most easily understood when one views the quadratic forms as quadratic spaces. So, if (V, q1) and (W, q2) are quadratic spaces, with V,W vector spaces, then the tensor product is a quadratic form q on the tensor product of vector spaces VW.

It is defined in such a way that for v \otimes w \in V \otimes W we have q(v \otimes w) = q_1(v)q_2(w). In particular, if we have diagonalizations of our quadratic forms (which is always possible when the characteristic is not 2) such that

q_1 \cong \langle a_1, ... , a_n \rangle
q_2 \cong \langle b_1, ... , b_m \rangle

then the tensor product has diagonalization

q_1 \otimes q_2 = q \cong \langle a_1b_1, a_1b_2, ... a_1b_m, a_2b_1, ... , a_2b_m , ... , a_nb_1, ... a_nb_m \rangle.