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.[1] If R is a commutative ring where 2 is invertible, and if and are two quadratic spaces over R, then their tensor product is the quadratic space whose underlying R-module is the tensor product of R-modules and whose quadratic form is the quadratic form associated to the tensor product of the bilinear forms associated to and .

In particular, the form satisfies

(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.,

then the tensor product has diagonalization

References

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  1. ^ Kitaoka, Yoshiyuki. "Tensor products of positive definite quadratic forms IV". Cambridge University Press. Retrieved February 12, 2024.