From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 20 Oct 2022 18:25:31 +0200
Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

---
 source/know/concept/gram-schmidt-method/index.md | 18 +++++++++---------
 1 file changed, 9 insertions(+), 9 deletions(-)

(limited to 'source/know/concept/gram-schmidt-method')

diff --git a/source/know/concept/gram-schmidt-method/index.md b/source/know/concept/gram-schmidt-method/index.md
index 374b169..70ad512 100644
--- a/source/know/concept/gram-schmidt-method/index.md
+++ b/source/know/concept/gram-schmidt-method/index.md
@@ -9,36 +9,36 @@ 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/),
+$$\ket{V_1}, \ket{V_2}, ...$$ 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:
+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 $\ket{n_1}$:
+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
+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
+    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
+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 $\ket{V_{j+1}}$.
+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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