Gram-Schmidt Algorithm

A process to make convert a set into an Orthogonal Set

Procedure

• Let $\{{\beta }_1,\dots ,{\beta }_j\}$ be a Linearly Independent set of vectors
• Consider set of vectors $\{{\alpha }_1,\dots ,{\alpha }_n\}$

1. ${\alpha }_1={\beta }_1$
2. ${\alpha }_2={\beta }_2-\frac{⟨{\beta }_2|{\alpha }_1⟩}{⟨{\alpha }_1|{\alpha }_1⟩}{\alpha }_1$
3. …
4. ${\alpha }_j={\beta }_k-{\sum }_{i=1}^{k-1}\frac{⟨{\beta }_k|{\alpha }_i⟩}{⟨{\alpha }_i|{\alpha }_1⟩}{\alpha }_i$

Intuition

1. Pick a first vector to base everything off. This first vector is now a fundamental basis vector.
2. The second vector is projected down with the first vector, and that component is subtracted from the second vector. This second vector is now a fundamental basis vector
3. The third vector is projected down with second, that component is subtracted, projected down with first vector, that component is subtracted. Now the third vector is a fundamental basis vector
4. Repeat for all other vectors