Linear Algebra and Its Applications · Problem 3.4.5

Orthogonality — Gilbert Strang

Mathematics · householder-reflection

Problem

u

is a unit vector, show that

Q = I - 2uu^T

is a symmetric orthogonal matrix. (It is a reflection, also known as a Householder transformation.) Compute

Q

when

u^T = \begin{bmatrix} \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} \end{bmatrix}