 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2009/p/p4/b/solution

x \in S_{v}

, then

v^{T} x = 0

and

P x = 0

, so

H_{v} x = x

. For arbitrary

x

v^{T} P = v^{T}

v^{T} H_{v} x = v^{T} x - 2 v^{T} P x = - v^{T} x .

Thus the Householder map fixes the hyperplane orthogonal to

v

and reverses the component in its normal direction: it is a reflection, not merely an arbitrary orthogonal map.

 New to topics? Read the docs here!