# ML Wiki

## Orthogonal Matrices

### Orthonormal Vectors

Vectors $\mathbf q_1, \ ... \ , \mathbf q_n$ are orthonormal if they are orthogonal and unit vectors

• $\mathbf q_i \; \bot \; \mathbf q_j \ \forall i \ne j$ and
• $\mathbf q_i^T \mathbf q_j = 0$ if $i \ne j$ and $\mathbf q_i^T \mathbf q_j = 1$ otherwise
• these vectors make a good basis

### Orthogonal Matrix

• The second part of the definition: $\mathbf q_i^T \mathbf q_j = \begin{cases} 1 & \text{if } i \ne j \\ 0 & \text{if } i = j \end{cases}$