Tensor product of quadratic forms

In mathematics, the tensor product of quadratic forms is most easily understood when one views the quadratic forms as quadratic spaces. If R is a commutative ring where 2 is invertible, and if $(V_1,q_1)$ and $(V_2,q_2)$ are two quadratic spaces over R, then their tensor product $(V_1\otimes V_2,q_1\otimes q_2)$ is the quadratic space whose underlying R-module is the tensor product $V_1\otimes V_2$ of R-modules and whose quadratic form is the quadratic form associated to the tensor product of the bilinear forms associated to $q_1$ and $q_2$.

In particular, the form $q_1\otimes q_2$ satisfies

$(q_1\otimes q_2)(v_1\otimes v_2)=q_1(v_1)q_2(v_2)\mathrm{\forall }{v}_1\in V_1,v_2\in V_2$

(which does uniquely characterize it however). It follows from this that if the quadratic forms are diagonalizable (which is always possible if 2 is invertible in R), i.e.,

$q_1\cong ⟨a_1,\mathrm{...},a_n⟩$

$q_2\cong ⟨b_1,\mathrm{...},b_m⟩$

then the tensor product has diagonalization

$q_1\otimes q_2\cong ⟨a_1{b}_1,a_1{b}_2,\mathrm{...}{a}_1{b}_m,a_2{b}_1,\mathrm{...},a_2{b}_m,\mathrm{...},a_n{b}_1,\mathrm{...}{a}_n{b}_m⟩.$