 Solution

ID: past-exam-of-the-numerical-linear-algebra-and-approximation-course-of-the-university-of-oxford/2008/1/a/ii/solution

H = I - 2 \frac{v v ^{T}}{v ^{T} v}, v \neq = 0.

It reflects in the hyperplane perpendicular to

v

. Put

P = v v^{T} / (v^{T} v)

P^{T} = P

and

P^{2} = P

. Then

H^{T} H = (I - 2 P)^{2} = I - 4 P + 4 P^{2} = I

, proving that every Householder matrix is orthogonal.

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