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
q
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
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
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