 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-1/17/e/solution

The matrix is

P = I - p p^{T} = 0 - 1 0 - 1 00 001,

P x = (- x_{2}, - x_{1}, x_{3})^{T} .

Here

∥ p ∥^{2} = 2

, so

P = I - 2 \hat{p} \hat{p}^{T}

with

\hat{p} = p / 2

. This is a Householder transformation: reflection in the plane $x_{1} + x_{2} = 0$ . The component parallel to the normal vector

p

changes sign, while the perpendicular component is fixed.

 New to topics? Read the docs here!