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 
.
It is defined in such a way that for 
we have 
. In particular, if we have diagonalizations of our quadratic forms (which is always possible when the characteristic is not 2) such that
then the tensor product has diagonalization
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