---
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.