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% Gram-Schmidt method
# Gram-Schmidt method
Given a set of linearly independent non-orthonormal vectors
$\ket*{V_1}, \ket*{V_2}, ...$ from a Hilbert space, the **Gram-Schmidt method**
turns them into an orthonormal set $\ket*{n_1}, \ket*{n_2}, ...$ as follows:
1. Take the first vector $\ket*{V_1}$ and normalize it to get $\ket*{n_1}$:
$$\begin{aligned}
\ket*{n_1} = \frac{\ket*{V_1}}{\sqrt{\braket*{V_1}{V_1}}}
\end{aligned}$$
2. Begin loop. Take the next non-orthonormal vector $\ket*{V_j}$, and
subtract from it its projection onto every already-processed vector:
$$\begin{aligned}
\ket*{n_j'} = \ket*{V_j} - \ket*{n_1} \braket*{n_1}{V_j} - \ket*{n_2} \braket*{n_2}{V_j} - ... - \ket*{n_{j-1}} \braket*{n_{j-1}}{V_{j-1}}
\end{aligned}$$
This leaves only the part of $\ket*{V_j}$ which is orthogonal to
$\ket*{n_1}$, $\ket*{n_2}$, etc. This why the input vectors must be
linearly independent; otherwise $\ket{n_j'}$ may become zero at some
point.
3. Normalize the resulting ortho*gonal* vector $\ket*{n_j'}$ to make it
ortho*normal*:
$$\begin{aligned}
\ket*{n_j} = \frac{\ket*{n_j'}}{\sqrt{\braket*{n_j'}{n_j'}}}
\end{aligned}$$
4. Loop back to step 2, taking the next vector $\ket*{V_{j+1}}$.
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