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 $v\otimes w\in V\otimes W$ 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

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_{1}b_{1},a_{1}b_{2},...a_{1}b_{m},a_{2}b_{1},...,a_{2}b_{m},...,a_{n}b_{1},...a_{n}b_{m}\rangle .

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