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---
title: "Gram-Schmidt method"
sort_title: "Gram-Schmidt method"
date: 2021-02-22
categories:
- Mathematics
- Algorithms
layout: "concept"
---
Given a set of $$N$$ linearly independent vectors $$\ket{V_1}, \ket{V_2}, ...$$
from a [Hilbert space](/know/concept/hilbert-space/),
the **Gram-Schmidt method** is an algorithm that turns them
into an orthonormal set $$\ket{n_1}, \ket{n_2}, ...$$ as follows
(in [Dirac notation](/know/concept/dirac-notation/)):
1. Take the first vector $$\ket{V_1}$$ and normalize it to get $$\ket{n_1}$$:
$$\begin{aligned}
\ket{n_1}
= \frac{\ket{V_1}}{\sqrt{\inprod{V_1}{V_1}}}
\end{aligned}$$
2. Begin loop. Take the next input vector $$\ket{V_j}$$, and
subtract from it its projection onto every already-processed vector:
$$\begin{aligned}
\ket{g_j}
= \ket{V_j} - \ket{n_1} \inprod{n_1}{V_j} - \ket{n_2} \inprod{n_2}{V_j} - ... - \ket{n_{j-1}} \inprod{n_{j-1}}{V_j}
\end{aligned}$$
This leaves only the part of $$\ket{V_j}$$
that is orthogonal to all previous $$\ket{n_k}$$.
This why the input vectors must be linearly independent;
otherwise $$\ket{g_j}$$ could become zero.
On a computer, the resulting $$\ket{g_j}$$ will
not be perfectly orthogonal due to rounding errors.
The above description of step #2 is particularly bad.
A better approach is:
$$\begin{aligned}
\ket{g_j^{(1)}}
&= \ket{V_j} - \ket{n_1} \inprod{n_1}{V_j}
\\
\ket{g_j^{(2)}}
&= \ket{g_j^{(1)}} - \ket{n_2} \inprod{n_2}{g_j^{(1)}}
\\
\vdots
\\
\ket{g_j}
= \ket{g_j^{(j-1)}}
&= \ket{g_j^{(j-2)}} - \ket{n_{j-2}} \inprod{n_{j-2}}{g_j^{(j-2)}}
\end{aligned}$$
In other words, instead of projecting $$\ket{V_j}$$ directly onto all $$\ket{n_k}$$,
we instead project only the part of $$\ket{V_j}$$ that has already been made orthogonal
to all previous $$\ket{n_m}$$ with $$m < k$$.
This is known as the **modified Gram-Schmidt method**.
3. Normalize the resulting ortho*gonal* vector $$\ket{g_j}$$ to make it ortho*normal*:
$$\begin{aligned}
\ket{n_j}
= \frac{\ket{g_j}}{\sqrt{\inprod{g_j}{g_j}}}
\end{aligned}$$
4. Loop back to step 2, taking the next vector $$\ket{V_{j+1}}$$,
until all $$N$$ have been processed.
## References
1. R. Shankar,
*Principles of quantum mechanics*, 2nd edition,
Springer.
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