From 5aeb822fd8dc079dbf4f03694ec6f79ce235d48e Mon Sep 17 00:00:00 2001
From: mhjensen <morten.hjorth-jensen@fys.uio.no>
Date: Tue, 22 Oct 2019 15:54:52 +0200
Subject: [PATCH] added pca theorem proof

---
 doc/pub/DimRed/html/._DimRed-bs000.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs001.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs002.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs003.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs004.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs005.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs006.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs007.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs008.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs009.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs010.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs011.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs012.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs013.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs014.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs015.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs016.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs017.html      |   6 +-
 doc/pub/DimRed/html/._DimRed-bs018.html      |  20 ++++-
 doc/pub/DimRed/html/._DimRed-bs019.html      |  24 +++++-
 doc/pub/DimRed/html/._DimRed-bs020.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs021.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs022.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs023.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs024.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs025.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs026.html      |   4 +-
 doc/pub/DimRed/html/._DimRed-bs027.html      |   4 +-
 doc/pub/DimRed/html/DimRed-bs.html           |   4 +-
 doc/pub/DimRed/html/DimRed-reveal.html       |  46 +++++++++++-
 doc/pub/DimRed/html/DimRed-solarized.html    |  40 +++++++++-
 doc/pub/DimRed/html/DimRed.html              |  40 +++++++++-
 doc/pub/DimRed/ipynb/DimRed.ipynb            |  75 ++++++++++++++++++-
 doc/pub/DimRed/ipynb/ipynb-DimRed-src.tar.gz | Bin 191 -> 191 bytes
 doc/pub/DimRed/pdf/DimRed-minted.pdf         | Bin 245191 -> 279641 bytes
 doc/src/DimRed/DimRed.do.txt                 |  41 ++++++++--
 36 files changed, 319 insertions(+), 77 deletions(-)

diff --git a/doc/pub/DimRed/html/._DimRed-bs000.html b/doc/pub/DimRed/html/._DimRed-bs000.html
index 3df86eb8f..91cab9210 100644
--- a/doc/pub/DimRed/html/._DimRed-bs000.html
+++ b/doc/pub/DimRed/html/._DimRed-bs000.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs001.html b/doc/pub/DimRed/html/._DimRed-bs001.html
index 6839f50c7..d27224c80 100644
--- a/doc/pub/DimRed/html/._DimRed-bs001.html
+++ b/doc/pub/DimRed/html/._DimRed-bs001.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs002.html b/doc/pub/DimRed/html/._DimRed-bs002.html
index 782040d1f..43db98f35 100644
--- a/doc/pub/DimRed/html/._DimRed-bs002.html
+++ b/doc/pub/DimRed/html/._DimRed-bs002.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs003.html b/doc/pub/DimRed/html/._DimRed-bs003.html
index b8f2304bf..bf6c5a14b 100644
--- a/doc/pub/DimRed/html/._DimRed-bs003.html
+++ b/doc/pub/DimRed/html/._DimRed-bs003.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs004.html b/doc/pub/DimRed/html/._DimRed-bs004.html
index 254726105..3301f1ff5 100644
--- a/doc/pub/DimRed/html/._DimRed-bs004.html
+++ b/doc/pub/DimRed/html/._DimRed-bs004.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs005.html b/doc/pub/DimRed/html/._DimRed-bs005.html
index 80b6dd521..86a86103e 100644
--- a/doc/pub/DimRed/html/._DimRed-bs005.html
+++ b/doc/pub/DimRed/html/._DimRed-bs005.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs006.html b/doc/pub/DimRed/html/._DimRed-bs006.html
index 4bde00dfe..670708858 100644
--- a/doc/pub/DimRed/html/._DimRed-bs006.html
+++ b/doc/pub/DimRed/html/._DimRed-bs006.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs007.html b/doc/pub/DimRed/html/._DimRed-bs007.html
index 4c99798ec..df6017a07 100644
--- a/doc/pub/DimRed/html/._DimRed-bs007.html
+++ b/doc/pub/DimRed/html/._DimRed-bs007.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs008.html b/doc/pub/DimRed/html/._DimRed-bs008.html
index 29b273d42..61a2e91bc 100644
--- a/doc/pub/DimRed/html/._DimRed-bs008.html
+++ b/doc/pub/DimRed/html/._DimRed-bs008.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs009.html b/doc/pub/DimRed/html/._DimRed-bs009.html
index 0633ff21e..2c062364d 100644
--- a/doc/pub/DimRed/html/._DimRed-bs009.html
+++ b/doc/pub/DimRed/html/._DimRed-bs009.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs010.html b/doc/pub/DimRed/html/._DimRed-bs010.html
index d35499f33..4dfdd3345 100644
--- a/doc/pub/DimRed/html/._DimRed-bs010.html
+++ b/doc/pub/DimRed/html/._DimRed-bs010.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs011.html b/doc/pub/DimRed/html/._DimRed-bs011.html
index 651f77edc..e13c5c7c3 100644
--- a/doc/pub/DimRed/html/._DimRed-bs011.html
+++ b/doc/pub/DimRed/html/._DimRed-bs011.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs012.html b/doc/pub/DimRed/html/._DimRed-bs012.html
index 8a75b27c5..e1767f7a6 100644
--- a/doc/pub/DimRed/html/._DimRed-bs012.html
+++ b/doc/pub/DimRed/html/._DimRed-bs012.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs013.html b/doc/pub/DimRed/html/._DimRed-bs013.html
index aadce9ddc..2c7239f42 100644
--- a/doc/pub/DimRed/html/._DimRed-bs013.html
+++ b/doc/pub/DimRed/html/._DimRed-bs013.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs014.html b/doc/pub/DimRed/html/._DimRed-bs014.html
index 847ed1de2..d99b56d86 100644
--- a/doc/pub/DimRed/html/._DimRed-bs014.html
+++ b/doc/pub/DimRed/html/._DimRed-bs014.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs015.html b/doc/pub/DimRed/html/._DimRed-bs015.html
index 5a51c64a1..1c1a2f6f6 100644
--- a/doc/pub/DimRed/html/._DimRed-bs015.html
+++ b/doc/pub/DimRed/html/._DimRed-bs015.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs016.html b/doc/pub/DimRed/html/._DimRed-bs016.html
index ba3075c88..6c0b5dad2 100644
--- a/doc/pub/DimRed/html/._DimRed-bs016.html
+++ b/doc/pub/DimRed/html/._DimRed-bs016.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs017.html b/doc/pub/DimRed/html/._DimRed-bs017.html
index 91074e516..c263831ce 100644
--- a/doc/pub/DimRed/html/._DimRed-bs017.html
+++ b/doc/pub/DimRed/html/._DimRed-bs017.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
@@ -203,7 +203,7 @@ $$
 
 <ul>
 <li> Center the data by subtracting the mean value for each column. This leads to a new matrix \( \boldsymbol{X}\rightarrow \overline{\boldsymbol{X}} \).</li>
-<li> Compute then the covariance/correlation matrix $\mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T].</li>
+<li> Compute then the covariance/correlation matrix \( \mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T] \).</li>
 <li> Find the eigenpairs of \( \boldsymbol{C} \) with eigenvalues \( [\lambda_0,\lambda_1,\dots,\lambda_{p-1}] \) and eigenvectors \( [\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \).</li>
 <li> Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.</li>
 <li> Keep only those \( l \) eigenvalues larger than a selected threshold value, discarding thus \( p-l \) features since we expect small variations in the data here.</li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs018.html b/doc/pub/DimRed/html/._DimRed-bs018.html
index 0e27b1a8b..cdf76d830 100644
--- a/doc/pub/DimRed/html/._DimRed-bs018.html
+++ b/doc/pub/DimRed/html/._DimRed-bs018.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
@@ -182,6 +182,22 @@ MathJax.Hub.Config({
 
 <h2 id="___sec17" class="anchor">Classical PCA Theorem   </h2>
 
+<p>
+We assume now that we have a design matrix \( \boldsymbol{X} \) which has been centered as discussed above. For the sake of simplicity we skip the overline symbol. The matrix is defined in terms of the various column vectors \( [\boldsymbol{x}_0,\boldsymbol{x}_1,\dots, \boldsymbol{x}_{p-1}] \)
+each with dimension \( \boldsymbol{x}\in {\mathbb{R}}^{n} \).
+
+<p>
+We assume also that we have an orthogonal transformation \( \boldsymbol{W}\in {\mathbb{R}}^{p\times p} \).  We define the reconstruction error (which is similar to the mean squared error we have seen before) as
+$$
+J(\boldsymbol{W},\boldsymbol{Z}) = \frac{1}{p}\sum_i (\boldsymbol{x}_i - \overline{\boldsymbol{x}_i})^2,
+$$
+
+with \( \overline{\boldsymbol{x}_i} = \boldsymbol{W}\boldsymbol{z}_i \), where \( \boldsymbol{z}_i \) is a row vector with dimension \( {\mathbb{R}}^{n} \) of the matrix
+\( \boldsymbol{Z}\in {\mathbb{R}}^{p\times n} \).
+
+<p>
+The PCA theorem states that minimizing the above reconstruction error corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors \( l \) $\boldsymbol{z}_i$, with \( l < < p \), defined by the orthogonal projection of the data onto the columns spanned by they eigenvectors of the covariance(correlations matrix).
+
 <p>
 <p>
 <!-- navigation buttons at the bottom of the page -->
diff --git a/doc/pub/DimRed/html/._DimRed-bs019.html b/doc/pub/DimRed/html/._DimRed-bs019.html
index 57a96ad5b..313bb5b43 100644
--- a/doc/pub/DimRed/html/._DimRed-bs019.html
+++ b/doc/pub/DimRed/html/._DimRed-bs019.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
@@ -180,7 +180,25 @@ MathJax.Hub.Config({
 <a name="part0019"></a>
 <!-- !split -->
 
-<h2 id="___sec18" class="anchor">Prof of the PCA Theorem  </h2>
+<h2 id="___sec18" class="anchor">Proof of the PCA Theorem  </h2>
+
+<p>
+To show the PCA theorem let us start with the assumption that there is a vector \( \boldsymbol{w}_0 \) which corresponds to a solution which minimized the reconstruction error \( J \). This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of \( \boldsymbol{w}_0 \) and \( \boldsymbol{z}_0 \) as
+$$
+J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{p}\sum_i (\boldsymbol{x}_^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0),
+$$
+
+which we can rewrite due to the orthogonality of \( \boldsymbol{w}_i \) as
+$$
+J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{p}\sum_i (\boldsymbol{x}_^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2).
+$$
+
+Minimizing \( J \) with respect to the unknown parameters \( z_{0i} \) we obtain that
+$$
+z_{i0}=\boldsymbol{w}_0^T\boldsymbol{x}_i,
+$$
+
+where the vectors on the rhs are known.
 
 <p>
 <p>
diff --git a/doc/pub/DimRed/html/._DimRed-bs020.html b/doc/pub/DimRed/html/._DimRed-bs020.html
index 5c47f1973..11b293e38 100644
--- a/doc/pub/DimRed/html/._DimRed-bs020.html
+++ b/doc/pub/DimRed/html/._DimRed-bs020.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs021.html b/doc/pub/DimRed/html/._DimRed-bs021.html
index ede6d1c65..d5a312ff5 100644
--- a/doc/pub/DimRed/html/._DimRed-bs021.html
+++ b/doc/pub/DimRed/html/._DimRed-bs021.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs022.html b/doc/pub/DimRed/html/._DimRed-bs022.html
index 2b727e37e..2e2ec0180 100644
--- a/doc/pub/DimRed/html/._DimRed-bs022.html
+++ b/doc/pub/DimRed/html/._DimRed-bs022.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs023.html b/doc/pub/DimRed/html/._DimRed-bs023.html
index 72d6189cd..aefe56512 100644
--- a/doc/pub/DimRed/html/._DimRed-bs023.html
+++ b/doc/pub/DimRed/html/._DimRed-bs023.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs024.html b/doc/pub/DimRed/html/._DimRed-bs024.html
index 9c87e99a1..a86276aa9 100644
--- a/doc/pub/DimRed/html/._DimRed-bs024.html
+++ b/doc/pub/DimRed/html/._DimRed-bs024.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs025.html b/doc/pub/DimRed/html/._DimRed-bs025.html
index 8da2c7966..d3fe90508 100644
--- a/doc/pub/DimRed/html/._DimRed-bs025.html
+++ b/doc/pub/DimRed/html/._DimRed-bs025.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs026.html b/doc/pub/DimRed/html/._DimRed-bs026.html
index 52f5488e9..e15057617 100644
--- a/doc/pub/DimRed/html/._DimRed-bs026.html
+++ b/doc/pub/DimRed/html/._DimRed-bs026.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/._DimRed-bs027.html b/doc/pub/DimRed/html/._DimRed-bs027.html
index ca75ce89c..5d51e8360 100644
--- a/doc/pub/DimRed/html/._DimRed-bs027.html
+++ b/doc/pub/DimRed/html/._DimRed-bs027.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/DimRed-bs.html b/doc/pub/DimRed/html/DimRed-bs.html
index 3df86eb8f..91cab9210 100644
--- a/doc/pub/DimRed/html/DimRed-bs.html
+++ b/doc/pub/DimRed/html/DimRed-bs.html
@@ -90,7 +90,7 @@ Automatically generated HTML file from DocOnce source
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -155,7 +155,7 @@ MathJax.Hub.Config({
      <!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
-     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
+     <!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">Getting started with PCA</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Principal Component Analysis</a></li>
      <!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">PCA and scikit-learn</a></li>
diff --git a/doc/pub/DimRed/html/DimRed-reveal.html b/doc/pub/DimRed/html/DimRed-reveal.html
index 221d0834e..b2cdac307 100644
--- a/doc/pub/DimRed/html/DimRed-reveal.html
+++ b/doc/pub/DimRed/html/DimRed-reveal.html
@@ -977,7 +977,7 @@ $$
 
 <ul>
 <p><li> Center the data by subtracting the mean value for each column. This leads to a new matrix \( \boldsymbol{X}\rightarrow \overline{\boldsymbol{X}} \).</li>
-<p><li> Compute then the covariance/correlation matrix $\mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T].</li>
+<p><li> Compute then the covariance/correlation matrix \( \mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T] \).</li>
 <p><li> Find the eigenpairs of \( \boldsymbol{C} \) with eigenvalues \( [\lambda_0,\lambda_1,\dots,\lambda_{p-1}] \) and eigenvectors \( [\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \).</li>
 <p><li> Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.</li>
 <p><li> Keep only those \( l \) eigenvalues larger than a selected threshold value, discarding thus \( p-l \) features since we expect small variations in the data here.</li>
@@ -990,11 +990,53 @@ After this we ask ourselves how do we prove the link between the maximum varianc
 
 <section>
 <h2 id="___sec17">Classical PCA Theorem   </h2>
+
+<p>
+We assume now that we have a design matrix \( \boldsymbol{X} \) which has been centered as discussed above. For the sake of simplicity we skip the overline symbol. The matrix is defined in terms of the various column vectors \( [\boldsymbol{x}_0,\boldsymbol{x}_1,\dots, \boldsymbol{x}_{p-1}] \)
+each with dimension \( \boldsymbol{x}\in {\mathbb{R}}^{n} \).
+
+<p>
+We assume also that we have an orthogonal transformation \( \boldsymbol{W}\in {\mathbb{R}}^{p\times p} \).  We define the reconstruction error (which is similar to the mean squared error we have seen before) as
+<p>&nbsp;<br>
+$$
+J(\boldsymbol{W},\boldsymbol{Z}) = \frac{1}{p}\sum_i (\boldsymbol{x}_i - \overline{\boldsymbol{x}_i})^2,
+$$
+<p>&nbsp;<br>
+
+with \( \overline{\boldsymbol{x}_i} = \boldsymbol{W}\boldsymbol{z}_i \), where \( \boldsymbol{z}_i \) is a row vector with dimension \( {\mathbb{R}}^{n} \) of the matrix
+\( \boldsymbol{Z}\in {\mathbb{R}}^{p\times n} \).
+
+<p>
+The PCA theorem states that minimizing the above reconstruction error corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors \( l \) $\boldsymbol{z}_i$, with \( l < < p \), defined by the orthogonal projection of the data onto the columns spanned by they eigenvectors of the covariance(correlations matrix).
 </section>
 
 
 <section>
-<h2 id="___sec18">Prof of the PCA Theorem  </h2>
+<h2 id="___sec18">Proof of the PCA Theorem  </h2>
+
+<p>
+To show the PCA theorem let us start with the assumption that there is a vector \( \boldsymbol{w}_0 \) which corresponds to a solution which minimized the reconstruction error \( J \). This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of \( \boldsymbol{w}_0 \) and \( \boldsymbol{z}_0 \) as
+<p>&nbsp;<br>
+$$
+J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{p}\sum_i (\boldsymbol{x}_^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0),
+$$
+<p>&nbsp;<br>
+
+which we can rewrite due to the orthogonality of \( \boldsymbol{w}_i \) as
+<p>&nbsp;<br>
+$$
+J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{p}\sum_i (\boldsymbol{x}_^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2).
+$$
+<p>&nbsp;<br>
+
+Minimizing \( J \) with respect to the unknown parameters \( z_{0i} \) we obtain that
+<p>&nbsp;<br>
+$$
+z_{i0}=\boldsymbol{w}_0^T\boldsymbol{x}_i,
+$$
+<p>&nbsp;<br>
+
+where the vectors on the rhs are known.
 </section>
 
 
diff --git a/doc/pub/DimRed/html/DimRed-solarized.html b/doc/pub/DimRed/html/DimRed-solarized.html
index 006dc5c14..908e6efc9 100644
--- a/doc/pub/DimRed/html/DimRed-solarized.html
+++ b/doc/pub/DimRed/html/DimRed-solarized.html
@@ -110,7 +110,7 @@ div { text-align: justify; text-justify: inter-word; }
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -942,7 +942,7 @@ $$
 
 <ul>
 <li> Center the data by subtracting the mean value for each column. This leads to a new matrix \( \boldsymbol{X}\rightarrow \overline{\boldsymbol{X}} \).</li>
-<li> Compute then the covariance/correlation matrix $\mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T].</li>
+<li> Compute then the covariance/correlation matrix \( \mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T] \).</li>
 <li> Find the eigenpairs of \( \boldsymbol{C} \) with eigenvalues \( [\lambda_0,\lambda_1,\dots,\lambda_{p-1}] \) and eigenvectors \( [\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \).</li>
 <li> Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.</li>
 <li> Keep only those \( l \) eigenvalues larger than a selected threshold value, discarding thus \( p-l \) features since we expect small variations in the data here.</li>
@@ -955,10 +955,44 @@ After this we ask ourselves how do we prove the link between the maximum varianc
 
 <h2 id="___sec17">Classical PCA Theorem   </h2>
 
+<p>
+We assume now that we have a design matrix \( \boldsymbol{X} \) which has been centered as discussed above. For the sake of simplicity we skip the overline symbol. The matrix is defined in terms of the various column vectors \( [\boldsymbol{x}_0,\boldsymbol{x}_1,\dots, \boldsymbol{x}_{p-1}] \)
+each with dimension \( \boldsymbol{x}\in {\mathbb{R}}^{n} \).
+
+<p>
+We assume also that we have an orthogonal transformation \( \boldsymbol{W}\in {\mathbb{R}}^{p\times p} \).  We define the reconstruction error (which is similar to the mean squared error we have seen before) as
+$$
+J(\boldsymbol{W},\boldsymbol{Z}) = \frac{1}{p}\sum_i (\boldsymbol{x}_i - \overline{\boldsymbol{x}_i})^2,
+$$
+
+with \( \overline{\boldsymbol{x}_i} = \boldsymbol{W}\boldsymbol{z}_i \), where \( \boldsymbol{z}_i \) is a row vector with dimension \( {\mathbb{R}}^{n} \) of the matrix
+\( \boldsymbol{Z}\in {\mathbb{R}}^{p\times n} \).
+
+<p>
+The PCA theorem states that minimizing the above reconstruction error corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors \( l \) $\boldsymbol{z}_i$, with \( l < < p \), defined by the orthogonal projection of the data onto the columns spanned by they eigenvectors of the covariance(correlations matrix).
+
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec18">Prof of the PCA Theorem  </h2>
+<h2 id="___sec18">Proof of the PCA Theorem  </h2>
+
+<p>
+To show the PCA theorem let us start with the assumption that there is a vector \( \boldsymbol{w}_0 \) which corresponds to a solution which minimized the reconstruction error \( J \). This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of \( \boldsymbol{w}_0 \) and \( \boldsymbol{z}_0 \) as
+$$
+J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{p}\sum_i (\boldsymbol{x}_^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0),
+$$
+
+which we can rewrite due to the orthogonality of \( \boldsymbol{w}_i \) as
+$$
+J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{p}\sum_i (\boldsymbol{x}_^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2).
+$$
+
+Minimizing \( J \) with respect to the unknown parameters \( z_{0i} \) we obtain that
+$$
+z_{i0}=\boldsymbol{w}_0^T\boldsymbol{x}_i,
+$$
+
+where the vectors on the rhs are known.
 
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
diff --git a/doc/pub/DimRed/html/DimRed.html b/doc/pub/DimRed/html/DimRed.html
index f6e2efdb3..d9d23455b 100644
--- a/doc/pub/DimRed/html/DimRed.html
+++ b/doc/pub/DimRed/html/DimRed.html
@@ -115,7 +115,7 @@ div { text-align: justify; text-justify: inter-word; }
               ('Towards the PCA theorem', 2, None, '___sec15'),
               ('The Algorithm before the Theorem', 2, None, '___sec16'),
               ('Classical PCA Theorem', 2, None, '___sec17'),
-              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Proof of the PCA Theorem', 2, None, '___sec18'),
               ('Getting started with PCA', 2, None, '___sec19'),
               ('Principal Component Analysis', 2, None, '___sec20'),
               ('PCA and scikit-learn', 2, None, '___sec21'),
@@ -947,7 +947,7 @@ $$
 
 <ul>
 <li> Center the data by subtracting the mean value for each column. This leads to a new matrix \( \boldsymbol{X}\rightarrow \overline{\boldsymbol{X}} \).</li>
-<li> Compute then the covariance/correlation matrix $\mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T].</li>
+<li> Compute then the covariance/correlation matrix \( \mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T] \).</li>
 <li> Find the eigenpairs of \( \boldsymbol{C} \) with eigenvalues \( [\lambda_0,\lambda_1,\dots,\lambda_{p-1}] \) and eigenvectors \( [\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \).</li>
 <li> Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.</li>
 <li> Keep only those \( l \) eigenvalues larger than a selected threshold value, discarding thus \( p-l \) features since we expect small variations in the data here.</li>
@@ -960,10 +960,44 @@ After this we ask ourselves how do we prove the link between the maximum varianc
 
 <h2 id="___sec17">Classical PCA Theorem   </h2>
 
+<p>
+We assume now that we have a design matrix \( \boldsymbol{X} \) which has been centered as discussed above. For the sake of simplicity we skip the overline symbol. The matrix is defined in terms of the various column vectors \( [\boldsymbol{x}_0,\boldsymbol{x}_1,\dots, \boldsymbol{x}_{p-1}] \)
+each with dimension \( \boldsymbol{x}\in {\mathbb{R}}^{n} \).
+
+<p>
+We assume also that we have an orthogonal transformation \( \boldsymbol{W}\in {\mathbb{R}}^{p\times p} \).  We define the reconstruction error (which is similar to the mean squared error we have seen before) as
+$$
+J(\boldsymbol{W},\boldsymbol{Z}) = \frac{1}{p}\sum_i (\boldsymbol{x}_i - \overline{\boldsymbol{x}_i})^2,
+$$
+
+with \( \overline{\boldsymbol{x}_i} = \boldsymbol{W}\boldsymbol{z}_i \), where \( \boldsymbol{z}_i \) is a row vector with dimension \( {\mathbb{R}}^{n} \) of the matrix
+\( \boldsymbol{Z}\in {\mathbb{R}}^{p\times n} \).
+
+<p>
+The PCA theorem states that minimizing the above reconstruction error corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors \( l \) $\boldsymbol{z}_i$, with \( l < < p \), defined by the orthogonal projection of the data onto the columns spanned by they eigenvectors of the covariance(correlations matrix).
+
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 
-<h2 id="___sec18">Prof of the PCA Theorem  </h2>
+<h2 id="___sec18">Proof of the PCA Theorem  </h2>
+
+<p>
+To show the PCA theorem let us start with the assumption that there is a vector \( \boldsymbol{w}_0 \) which corresponds to a solution which minimized the reconstruction error \( J \). This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of \( \boldsymbol{w}_0 \) and \( \boldsymbol{z}_0 \) as
+$$
+J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{p}\sum_i (\boldsymbol{x}_^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0),
+$$
+
+which we can rewrite due to the orthogonality of \( \boldsymbol{w}_i \) as
+$$
+J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{p}\sum_i (\boldsymbol{x}_^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2).
+$$
+
+Minimizing \( J \) with respect to the unknown parameters \( z_{0i} \) we obtain that
+$$
+z_{i0}=\boldsymbol{w}_0^T\boldsymbol{x}_i,
+$$
+
+where the vectors on the rhs are known.
 
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
diff --git a/doc/pub/DimRed/ipynb/DimRed.ipynb b/doc/pub/DimRed/ipynb/DimRed.ipynb
index 538fd7b6f..dfc5ebe65 100644
--- a/doc/pub/DimRed/ipynb/DimRed.ipynb
+++ b/doc/pub/DimRed/ipynb/DimRed.ipynb
@@ -1041,7 +1041,7 @@
    "source": [
     "* Center the data by subtracting the mean value for each column. This leads to a new matrix $\\boldsymbol{X}\\rightarrow \\overline{\\boldsymbol{X}}$.\n",
     "\n",
-    "* Compute then the covariance/correlation matrix $\\mathbb{E}[\\overline{\\boldsymbol{X}}\\overline{\\boldsymbol{X}}^T].\n",
+    "* Compute then the covariance/correlation matrix $\\mathbb{E}[\\overline{\\boldsymbol{X}}\\overline{\\boldsymbol{X}}^T]$.\n",
     "\n",
     "* Find the eigenpairs of $\\boldsymbol{C}$ with eigenvalues $[\\lambda_0,\\lambda_1,\\dots,\\lambda_{p-1}]$ and eigenvectors $[\\boldsymbol{s}_0,\\boldsymbol{s}_1,\\dots,\\boldsymbol{s}_{p-1}]$.\n",
     "\n",
@@ -1053,14 +1053,83 @@
     "\n",
     "## Classical PCA Theorem\n",
     "\n",
+    "We assume now that we have a design matrix $\\boldsymbol{X}$ which has been centered as discussed above. For the sake of simplicity we skip the overline symbol. The matrix is defined in terms of the various column vectors $[\\boldsymbol{x}_0,\\boldsymbol{x}_1,\\dots, \\boldsymbol{x}_{p-1}]$\n",
+    "each with dimension $\\boldsymbol{x}\\in {\\mathbb{R}}^{n}$.\n",
     "\n",
+    "We assume also that we have an orthogonal transformation $\\boldsymbol{W}\\in {\\mathbb{R}}^{p\\times p}$.  We define the reconstruction error (which is similar to the mean squared error we have seen before) as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "J(\\boldsymbol{W},\\boldsymbol{Z}) = \\frac{1}{p}\\sum_i (\\boldsymbol{x}_i - \\overline{\\boldsymbol{x}_i})^2,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "with $\\overline{\\boldsymbol{x}_i} = \\boldsymbol{W}\\boldsymbol{z}_i$, where $\\boldsymbol{z}_i$ is a row vector with dimension ${\\mathbb{R}}^{n}$ of the matrix\n",
+    "$\\boldsymbol{Z}\\in {\\mathbb{R}}^{p\\times n}$.  \n",
     "\n",
-    "## Prof of the PCA Theorem\n",
-    "\n",
+    "The PCA theorem states that minimizing the above reconstruction error corresponds to setting $\\boldsymbol{W}=\\boldsymbol{S}$, the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors $l$ $\\boldsymbol{z}_i$, with $l << p$, defined by the orthogonal projection of the data onto the columns spanned by they eigenvectors of the covariance(correlations matrix).\n",
     "\n",
     "\n",
     "\n",
+    "## Proof of the PCA Theorem\n",
     "\n",
+    "To show the PCA theorem let us start with the assumption that there is a vector $\\boldsymbol{w}_0$ which corresponds to a solution which minimized the reconstruction error $J$. This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of $\\boldsymbol{w}_0$ and $\\boldsymbol{z}_0$ as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "J(\\boldsymbol{w}_0,\\boldsymbol{z}_0)= \\frac{1}{p}\\sum_i (\\boldsymbol{x}_i - z_{i0}\\boldsymbol{w}_0)^2=\\frac{1}{p}\\sum_i (\\boldsymbol{x}_^T\\boldsymbol{x}_i - 2z_{i0}\\boldsymbol{w}_0^T\\boldsymbol{x}_i+z_{i0}^2\\boldsymbol{w}_0^T\\boldsymbol{w}_0),\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "which we can rewrite due to the orthogonality of $\\boldsymbol{w}_i$ as"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "J(\\boldsymbol{w}_0,\\boldsymbol{z}_0)=\\frac{1}{p}\\sum_i (\\boldsymbol{x}_^T\\boldsymbol{x}_i - 2z_{i0}\\boldsymbol{w}_0^T\\boldsymbol{x}_i+z_{i0}^2).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Minimizing $J$ with respect to the unknown parameters $z_{0i}$ we obtain that"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "z_{i0}=\\boldsymbol{w}_0^T\\boldsymbol{x}_i,\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "where the vectors on the rhs are known. \n",
     "\n",
     "## Getting started with PCA"
    ]
diff --git a/doc/pub/DimRed/ipynb/ipynb-DimRed-src.tar.gz b/doc/pub/DimRed/ipynb/ipynb-DimRed-src.tar.gz
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diff --git a/doc/src/DimRed/DimRed.do.txt b/doc/src/DimRed/DimRed.do.txt
index a7124c982..301664605 100644
--- a/doc/src/DimRed/DimRed.do.txt
+++ b/doc/src/DimRed/DimRed.do.txt
@@ -734,7 +734,7 @@ x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
 \]
 !et
 * Center the data by subtracting the mean value for each column. This leads to a new matrix $\bm{X}\rightarrow \overline{\bm{X}}$.
-* Compute then the covariance/correlation matrix $\mathbb{E}[\overline{\bm{X}}\overline{\bm{X}}^T].
+* Compute then the covariance/correlation matrix $\mathbb{E}[\overline{\bm{X}}\overline{\bm{X}}^T]$.
 * Find the eigenpairs of $\bm{C}$ with eigenvalues $[\lambda_0,\lambda_1,\dots,\lambda_{p-1}]$ and eigenvectors $[\bm{s}_0,\bm{s}_1,\dots,\bm{s}_{p-1}]$.
 * Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.
 * Keep only those $l$ eigenvalues larger than a selected threshold value, discarding thus $p-l$ features since we expect small variations in the data here.
@@ -744,15 +744,44 @@ After this we ask ourselves how do we prove the link between the maximum varianc
 !split
 ===== Classical PCA Theorem   =====
 
+We assume now that we have a design matrix $\bm{X}$ which has been centered as discussed above. For the sake of simplicity we skip the overline symbol. The matrix is defined in terms of the various column vectors $[\bm{x}_0,\bm{x}_1,\dots, \bm{x}_{p-1}]$
+each with dimension $\bm{x}\in {\mathbb{R}}^{n}$.
+
+We assume also that we have an orthogonal transformation $\bm{W}\in {\mathbb{R}}^{p\times p}$.  We define the reconstruction error (which is similar to the mean squared error we have seen before) as
+!bt
+\[
+J(\bm{W},\bm{Z}) = \frac{1}{p}\sum_i (\bm{x}_i - \overline{\bm{x}_i})^2,
+\]
+!et
+with $\overline{\bm{x}_i} = \bm{W}\bm{z}_i$, where $\bm{z}_i$ is a row vector with dimension ${\mathbb{R}}^{n}$ of the matrix
+$\bm{Z}\in {\mathbb{R}}^{p\times n}$.  
+
+The PCA theorem states that minimizing the above reconstruction error corresponds to setting $\bm{W}=\bm{S}$, the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors $l$ $\bm{z}_i$, with $l << p$, defined by the orthogonal projection of the data onto the columns spanned by they eigenvectors of the covariance(correlations matrix).
+
 
 
 !split
-===== Prof of the PCA Theorem  =====
-
-
-
-
+===== Proof of the PCA Theorem  =====
 
+To show the PCA theorem let us start with the assumption that there is a vector $\bm{w}_0$ which corresponds to a solution which minimized the reconstruction error $J$. This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of $\bm{w}_0$ and $\bm{z}_0$ as
+!bt
+\[
+J(\bm{w}_0,\bm{z}_0)= \frac{1}{p}\sum_i (\bm{x}_i - z_{i0}\bm{w}_0)^2=\frac{1}{p}\sum_i (\bm{x}_^T\bm{x}_i - 2z_{i0}\bm{w}_0^T\bm{x}_i+z_{i0}^2\bm{w}_0^T\bm{w}_0),
+\]
+!et
+which we can rewrite due to the orthogonality of $\bm{w}_i$ as
+!bt
+\[
+J(\bm{w}_0,\bm{z}_0)=\frac{1}{p}\sum_i (\bm{x}_^T\bm{x}_i - 2z_{i0}\bm{w}_0^T\bm{x}_i+z_{i0}^2).
+\]
+!et
+Minimizing $J$ with respect to the unknown parameters $z_{0i}$ we obtain that
+!bt
+\[
+z_{i0}=\bm{w}_0^T\bm{x}_i,
+\]
+!et
+where the vectors on the rhs are known. 
 
 !split
 ===== Getting started with PCA =====