Gram-Schmidt Orthonormalisation Process

Algorithm: Gram-Schmidt Orthonormalisation Process

Every basis $B=\{{\mathbf{b}}_1,\cdots ,{\mathbf{b}}_n\}$ of a finite-dimensional inner product space $(V,F,+,\cdot )$ can be turned into an orthonormal basis $B^{\prime }=\{{\mathbf{b}}_1^{\prime },\cdots ,{\mathbf{b}}_n^{\prime }\}$ through the following process:

1. The first vector ${\mathbf{b}}_1^{\prime }$ in $B^{\prime }$ is simply the normalised version of ${\mathbf{b}}_1$.

$${\mathbf{b}}_1^{\prime }\leftarrow \frac{1}{||{\mathbf{b}}_1||}{\mathbf{b}}_1$$

2. We construct the $k$-th element ${\mathbf{b}}_k^{\prime }$ of $B^{\prime }$ so that it is orthogonal to all the resulting basis vectors $\{{\mathbf{b}}_1^{\prime },\cdots ,{\mathbf{b}}_{k-1}^{\prime }\}$ before it and so that its canonical norm is one.

• Firstly, we ensure that ${\mathbf{b}}_k^{\prime }$ is orthogonal to all $\{{\mathbf{b}}_1^{\prime },\cdots ,{\mathbf{b}}_{k-1}^{\prime }\}$ by assigning it the following value.

$${\mathbf{b}}_k^{\prime }\leftarrow {\mathbf{b}}_k-\sum _{i=1}^{k-1}⟨{\mathbf{b}}_k,{\mathbf{b}}_i^{\prime }⟩{\mathbf{b}}_i^{\prime }$$

• We then normalise ${\mathbf{b}}_k^{\prime }$ to ensure that its norm is one.

$${\mathbf{b}}_k^{\prime }\leftarrow \frac{1}{||{\mathbf{b}}_k^{\prime }||}{\mathbf{b}}_k^{\prime }$$