 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2011/p/p4/a/solution

For nonzero

w

H (w) = I - 2 \frac{w w ^{T}}{w ^{T} w} .

The rank-one matrix

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

is symmetric and satisfies

P^{2} = P

. Consequently

H^{T} = H

and

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

. Thus the matrix is symmetric and orthogonal; geometrically it reflects the component along

w

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