Tensor product of quadratic forms

Last updated March 02, 2024

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 $(V_{1},q_{1})$ 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})\quad \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 \langle a_{1},...,a_{n}\rangle

q_{2}\cong \langle b_{1},...,b_{m}\rangle

then the tensor product has diagonalization

q_{1}\otimes q_{2}\cong \langle a_{1}b_{1},a_{1}b_{2},...a_{1}b_{m},a_{2}b_{1},...,a_{2}b_{m},...,a_{n}b_{1},...a_{n}b_{m}\rangle .

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References

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

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

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