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Polarization identity

From Encyclopedia of Mathematics
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An identity relating a quadratic form to a bilinear form.

If 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
How to Cite This Entry:
Polarisation identity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polarisation_identity&oldid=37593