The Gram-Schmidt Algorithm
In the simplest review, the Gram-Schmidt Algorithm is shown in the following pattern for vectors u and v.
- $ \begin{align} \mathbf{u}_1 & = \mathbf{v}_1, \\ \mathbf{u}_2 & = \mathbf{v}_2-\mathrm{proj}_{\mathbf{u}_1}\,(\mathbf{v}_2), \\ \mathbf{u}_3 & = \mathbf{v}_3-\mathrm{proj}_{\mathbf{u}_1}\,(\mathbf{v}_3)-\mathrm{proj}_{\mathbf{u}_2}\,(\mathbf{v}_3), \\ & {}\ \ \vdots \\ \mathbf{u}_k & = \mathbf{v}_k-\sum_{j=1}^{k-1}\mathrm{proj}_{\mathbf{u}_j}\,(\mathbf{v}_k), \end{align} $
where
- $ \mathrm{proj}_{\mathbf{u}}\,(\mathbf{v}) = {\langle \mathbf{v}, \mathbf{u}\rangle\over\langle \mathbf{u}, \mathbf{u}\rangle}\mathbf{u} $.
