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+---
+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.