Orthonormal matrix

• categories: Linear algebra, Matrix

• Doesn’t change lengths $||Qx||=||x|{|}^2$

• No overflow when using $Q$

• $Q^T\times Q=Q\times Q^T=I$ for square matrix

• $Q^{-1}=Q^T$

• Eigenvectors are orthogonal

Examples :

1. Rotation $$\begin{gathered} \text{pmatrix}\cos \theta ,-\sin \theta \\ \sin \theta ,\cos \theta \end{gathered}$$
2. Reflection $$\begin{gathered} \text{pmatrix}\cos \theta ,\sin \theta \\ \sin \theta ,-\cos \theta \end{gathered}$$
3. Householder reflections: start with $U^{T}U=1=>H=I-2U^{T}U$ - family of symmetric orthonormal matrices
4. Haar Wavelet Matrix
5. Permutation Matrix