Orthogonal Matrix

A matrix whose columns are Orthonormal

$U$ is orthogonal $⟺U^{T}U=I_n$ $U\square ⟹U^T=U^{-1}$

Properties

Preserves Length

$||U\vec{x}||=||\vec{x}||$

Preserves Angles

$(U\vec{x})\cdot (U\vec{y})=\vec{x}\cdot \vec{y}$

Preserves Orthogonality

$(U\vec{x})\cdot (U\vec{y})=\vec{x}\cdot \vec{y}⟹(\vec{x}\cdot \vec{y}=0⟺(U\vec{x})\cdot (U\vec{y})=0)$