Prove that the inner product associated with a positive definite quadratic form q(x) is given by the polarization formula $⟨x,y⟩=1/2[q(x+y)−q(x)−q(y)]$.
How will I be able to do this problem?
Prove that the inner product associated with a positive definite quadratic form q(x) is given by the polarization formula $⟨x,y⟩=1/2[q(x+y)−q(x)−q(y)]$.
How will I be able to do this problem?
Since the association between quadratic form $q$ and innner product $\langle \cdot, \cdot \rangle$ is given by $q(x) = \langle x, x \rangle$, this is more-or-less direct calculation using the linearity of the inner product. In particular, note that
$q(x+y) = \langle x+y, x+y\rangle = \langle x, x\rangle + \langle x, y \rangle + \langle y, x \rangle + \langle y, y \rangle = q(x) + 2 \langle x,y\rangle + q(y),$ using linearity to expand $\langle x+y,x+y\rangle$ and symmetry to transform $\langle y,x\rangle$ to $\langle x, y\rangle$.