Let vectors $\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3$ construct Orthogonal vectors $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ and then Orthonormal vectors $\mathbf{q}_1, \mathbf{q}_2, \mathbf{q}_3$.
$$
\mathbf{v}_1 = \mathbf{u}_1 \rightarrow \mathbf{q}_1 = \frac{\mathbf{v}_1}{\|\mathbf{v}_1\|}
$$
$$
\mathbf{v}_2 = \mathbf{u}_2 - \frac{\mathbf{v}_1^T \mathbf{u}_2}{\mathbf{v}_1^T \mathbf{v}_1} \mathbf{v}_1 \rightarrow \mathbf{q}_2 = \frac{\mathbf{v}_2}{\|\mathbf{v}_2\|}
$$
$$
\mathbf{v}_3 = \mathbf{u}_3 - \frac{\mathbf{v}_1^T \mathbf{u}_3}{\mathbf{v}_1^T \mathbf{v}_1} \mathbf{v}_1 - \frac{\mathbf{v}_2^T \mathbf{u}_3}{\mathbf{v}_2^T \mathbf{v}_2} \mathbf{v}_2 \rightarrow \mathbf{q}_3 = \frac{\mathbf{v}_3}{\|\mathbf{v}_3\|}
$$
#Exercise
- Let vectors $\mathbf{u}_1 = \begin{bmatrix} 1 \ 1 \ 1 \end{bmatrix}$, $\mathbf{u}_2 = \begin{bmatrix} 1 \ 0 \ 2 \end{bmatrix}$. Find vectors $\mathbf{q}_1, \mathbf{q}_2$ and matrix $Q$.
Gauss Siedel Numerical Method