From 15bfb7730801809704c6561e20c5ca47627b2d79 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 21 Feb 2021 20:53:46 +0100
Subject: Add "Gram-Schmidt method"

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 latex/know/concept/gram-schmidt-method/source.md | 35 ++++++++++++++++++++++++
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+% Gram-Schmidt method
+
+
+# Gram-Schmidt method
+
+Given a set of linearly independent non-orthonormal vectors
+$\ket*{V_1}, \ket*{V_2}, ...$ from a 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 $\ket*{V_{j+1}}$.
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