Past exam of the numerical linear algebra and approximation course of the University of Oxford / 2008 / 1 / a / i

A real square orthogonal matrix $Q$ satisfies $\boxed{Q^{T}Q=Q{Q}^T=I}$. Equivalently its columns form an orthonormal basis. In particular it preserves the Euclidean norm, since $\Vert Qx{\Vert }_2^2=x^T{Q}^{T}Qx=\Vert x{\Vert }_2^2$.