Orthogonal Matrices

An $m\times n,m\ge n$ matrix has orthonormal columns $⟺U^{T}U=I_n$ Notice, $U^{T}U=I_n$ implies that $U^T=U^{-1}$

Properties

• $||U\vec{x}||=||\vec{x}||$
• $(U\vec{x})\cdot (U\vec{y})=\vec{x}\cdot \vec{y}$
• $(U\vec{x})\cdot (U\vec{y})=0⟺\vec{x}\cdot \vec{y}=0$
• $U^{-1}=U^T$
• $\det A=1\text{ or }-1$

$U^T=U^{-1}$