# Tensor product of quadratic forms

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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 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 follow from this that if we have diagonalizations of our quadratic forms (which is always possible if is invertible) such that

then the tensor product has diagonalization

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