From 2803531f74adf72de2f566d455b05186f9f03ea6 Mon Sep 17 00:00:00 2001
From: Trung Trinh <trung.trinh@aalto.fi>
Date: Sat, 4 May 2024 18:12:08 +0300
Subject: [PATCH] test

---
 index.md | 9 ++++++---
 1 file changed, 6 insertions(+), 3 deletions(-)

diff --git a/index.md b/index.md
index 7504059..afea119 100644
--- a/index.md
+++ b/index.md
@@ -63,7 +63,10 @@ Each <span class="my_blue">lengthscale</span> is inversely proportional to the s
 
 </div>
 
-<ul>
-<li> Use PCA to get the <span class="my_blue">eigenvalues</span> and <span class="my_red">eigenvectors</span> of the training data: \\(\\{ {\color{red}\mathbf{u}\_d},{\color[RGB]{68,114,196}\lambda\_d}\\}\_{d=1}^D\\)
+- Use PCA to get the <span class="my_blue">eigenvalues</span> and <span class="my_red">eigenvectors</span> of the training data: \\(\\{ {\color{red}\mathbf{u}\_d},{\color[RGB]{68,114,196}\lambda\_d}\\}\_{d=1}^D\\)
+- Define the base kernel:
 
-</ul>
\ No newline at end of file
+<img src="./assets/pca_kernel.svg" alt="drawing" width="80%" max-width="800px">
+
+    - \\( {\color{red}U} = \begin{bmatrix} {\color{red}\mathbf{u}\_1} & {\color{red}\mathbf{u}\_2} & \cdots & {\color{red}\mathbf{u}\_D} \end{bmatrix} \\) is a matrix containing the <span class="my_red">eigenvectors</span> as columns.
+    - \\( {\color[RGB]{68,114,196}\boldsymbol{\Sigma}^{-1}\_{\alpha}} = (1-\alpha)\mathbf{I} + \alpha {\color[RGB]{68,114,196}\boldsymbol{\Lambda} } \\) where \\( \color[RGB]{68,114,196}\boldsymbol{\Lambda} \\) is a diagonal matrix containing the <span class="my_blue">eigenvalues</span>