Tensor product of quadratic forms

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In mathematics, the tensor product of quadratic forms is most easily understood when one views the quadratic forms as quadratic spaces.[1] If R is a commutative ring where 2 is invertible, and if (V1,q1) and (V2,q2) are two quadratic spaces over R, then their tensor product (V1⊗V2,q1⊗q2) is the quadratic space whose underlying R-module is the tensor product V1⊗V2 of R-modules and whose quadratic form is the quadratic form associated to the tensor product of the bilinear forms associated to q1 and q2.

In particular, the form q1⊗q2 satisfies

(q1⊗q2)(v1⊗v2)=q1(v1)q2(v2)∀v1∈V1, v2∈V2

(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.,

q1≅⟨a1,...,an⟩
q2≅⟨b1,...,bm⟩

then the tensor product has diagonalization

q1⊗q2≅⟨a1b1,a1b2,...a1bm,a2b1,...,a2bm,...,anb1,...anbm⟩.

## References

1. ↑ Kitaoka, Yoshiyuki (1979). "Tensor products of positive definite quadratic forms IV". Nagoya Mathematical Journal (Cambridge University Press) 73: 149–156. doi:10.1017/S0027763000018365. https://www.cambridge.org/core/journals/nagoya-mathematical-journal/article/tensor-products-of-positive-definite-quadratic-forms-iv/1F360E61EDA5EBAA4C86C64F987C82D1. Retrieved February 12, 2024.

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