Polarization identity

An identity relating a quadratic form to a bilinear form.

If $q$ is a quadratic form on a vector space $V$ over a field of characteristic not equal to $2$, or more generally, a module over a ring in which $2$ is invertible, then defining $b$ by $$ b(x,y) = \frac12 ( q(x+y) - q(x) - q(y) ) $$ or $$ b(x,y) = \frac14 ( q(x+y) - q(x-y) ) $$ yields a symmetric bilinear form on $V$ such that $q(x) = b(x,x)$.

Similarly, if $q$ is a quadratic form over a complex vector space then $$ 4 b(x,y) = q(x+y) - q(x-y) + i (q(x + iy) - q(x-iy)) $$ defines a Hermitian form.

More generally there is an identity relating a homogenous form $F$ to a multilinear form $M$ over a field of characteristic zero $$ M(x_1,\ldots,x_n) = \frac{1}{k!} \sum_{ \emptyset \neq I \subseteq \{1,\ldots,n\} } (-1)^{n-|I|} F\left({\sum_{i \in I} x_i}\right) \ . $$

References

• Körner, T.W. A Companion to Analysis: A Second First and First Second Course in Analysis American Mathematical Soc. (2004) ISBN 0-8218-3447-9
• Landsberg, J.M. Tensors: Geometry and Applications American Mathematical Soc. (2011) ISBN 0-8218-6907-8