Tensor product of quadratic forms

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 V ⊗ W.

It is defined in such a way that for v ⊗ w ∈ V ⊗ W we have q ( v ⊗ 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

then the tensor product has diagonalization