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, 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 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