Definitions

# Tensor product of quadratic forms

The is most easily understood when one views the quadratic forms as quadratic spaces. So, if (V, q_1) and (W, q_2) are quadratic spaces, which V,W vector spaces, then the tensor product is a quadratic form q on the tensor product of vector spaces $V otimes W$.

It is defined in such a way that for $v otimes w in V otimes W$ we have $q\left(v otimes w\right) = q_1\left(v\right)q_2\left(w\right)$. 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

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