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Date: Wed, 24 Feb 2021 09:47:22 +0100
Subject: Initial commit
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+---
+title: "Gram-Schmidt method"
+firstLetter: "G"
+publishDate: 2021-02-22
+categories:
+- Mathematics
+
+date: 2021-02-22T21:36:08+01:00
+draft: false
+markup: pandoc
+---
+
+# Gram-Schmidt method
+
+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 $\ket*{n_1}$:
+
+ $$\begin{aligned}
+ \ket*{n_1} = \frac{\ket*{V_1}}{\sqrt{\braket*{V_1}{V_1}}}
+ \end{aligned}$$
+
+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 $\ket*{V_{j+1}}$.
+
+If you are unfamiliar with this notation, take a look at [Dirac notation](/know/concept/dirac-notation/).
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