Positive Definite Proof
By Emma Terry •
Prove that the inner product associated with a positive definite quadratic form $q(x)$ is given by the polarization formula $\langle x, y\rangle = \frac{1}{2}[q(x+y) - q(x) - q(y)]$.
How will I be able to do this problem. I know in order to be a positive definite you need the following axioms to be verified: bilinearity, positivity and symmetric. Thus we need that $\langle x, y\rangle = x^TKy$ for $x,y \in R^n$ but how do I go on to apply that here?
$\endgroup$1 Answer
$\begingroup$Don't need to apply that, as you wrote, the mentioned axioms are to be verified. How is a pos.def.quadratic form defined? That's what you can use.
But first of all, observe/verify that for given inner product $\langle,\rangle$, with $q(x):=\langle x,x\rangle$ the polarization formula holds.
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