blob: 483dd71e80346d9d65d4cebd13561c9167c44894 (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
|
---
title: "Gram-Schmidt method"
date: 2021-02-22
categories:
- Mathematics
- Algorithms
layout: "concept"
---
Given a set of linearly independent non-orthonormal vectors
$\ket{V_1}, \ket{V_2}, ...$ from a [Hilbert space](/know/concept/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 $\ket{n_1}$:
$$\begin{aligned}
\ket{n_1} = \frac{\ket{V_1}}{\sqrt{\inprod{V_1}{V_1}}}
\end{aligned}$$
2. Begin loop. Take the next non-orthonormal vector $\ket{V_j}$, and
subtract from it its projection onto every already-processed vector:
$$\begin{aligned}
\ket{n_j'} = \ket{V_j} - \ket{n_1} \inprod{n_1}{V_j} - \ket{n_2} \inprod{n_2}{V_j} - ... - \ket{n_{j-1}} \inprod{n_{j-1}}{V_{j-1}}
\end{aligned}$$
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*:
$$\begin{aligned}
\ket{n_j} = \frac{\ket{n_j'}}{\sqrt{\inprod{n_j'}{n_j'}}}
\end{aligned}$$
4. Loop back to step 2, taking the next vector $\ket{V_{j+1}}$.
If you are unfamiliar with this notation, take a look at [Dirac notation](/know/concept/dirac-notation/).
## References
1. R. Shankar,
*Principles of quantum mechanics*, 2nd edition,
Springer.
|
