Tensor product of quadratic forms

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The tensor product of quadratic forms 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 $q(v \otimes w) = q_1(v)q_2(w)$ . In particular, if we have diagonalizations of our quadratic forms (which is always possible when the characteristic is not 2) such that