From 05c0090fd82e71fd1cf10951afe0eede6fd33196 Mon Sep 17 00:00:00 2001
From: mhjensen <morten.hjorth-jensen@fys.uio.no>
Date: Thu, 5 Sep 2019 13:42:18 +0200
Subject: [PATCH] Added material to slides

---
 .../Regression/html/._Regression-bs042.html   |   5 +++++
 .../Regression/html/._Regression-bs044.html   |   2 +-
 .../Regression/html/Regression-reveal.html    |   7 ++++++-
 .../Regression/html/Regression-solarized.html |   7 ++++++-
 doc/pub/Regression/html/Regression.html       |   7 ++++++-
 doc/pub/Regression/ipynb/Regression.ipynb     |   7 ++++++-
 .../ipynb/ipynb-Regression-src.tar.gz         | Bin 211 -> 212 bytes
 doc/pub/Regression/pdf/Regression-minted.pdf  | Bin 453646 -> 453864 bytes
 doc/src/Regression/Regression.do.txt          |   7 ++++++-
 9 files changed, 36 insertions(+), 6 deletions(-)

diff --git a/doc/pub/Regression/html/._Regression-bs042.html b/doc/pub/Regression/html/._Regression-bs042.html
index 5aed6efbe..525941a4c 100644
--- a/doc/pub/Regression/html/._Regression-bs042.html
+++ b/doc/pub/Regression/html/._Regression-bs042.html
@@ -407,9 +407,14 @@ MathJax.Hub.Config({
 <span style="color: #BA2121; font-style: italic">    numpy and scipy.linalg at the cost of being slower.</span>
 <span style="color: #BA2121; font-style: italic">    &#39;&#39;&#39;</span>
     U, s, VT <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>svd(A)
+<span style="color: #408080; font-style: italic">#    print(&#39;test U&#39;)</span>
+<span style="color: #408080; font-style: italic">#    print( (np.transpose(U) @ U - U @np.transpose(U)))</span>
+<span style="color: #408080; font-style: italic">#    print(&#39;test VT&#39;)</span>
+<span style="color: #408080; font-style: italic">#    print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))</span>
     <span style="color: #008000; font-weight: bold">print</span>(U)
     <span style="color: #008000; font-weight: bold">print</span>(s)
     <span style="color: #008000; font-weight: bold">print</span>(VT)
+
     D <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(U),<span style="color: #008000">len</span>(VT)))
     <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">0</span>,<span style="color: #008000">len</span>(VT)):
         D[i,i]<span style="color: #666666">=</span>s[i]
diff --git a/doc/pub/Regression/html/._Regression-bs044.html b/doc/pub/Regression/html/._Regression-bs044.html
index 932067660..3f96d2e2e 100644
--- a/doc/pub/Regression/html/._Regression-bs044.html
+++ b/doc/pub/Regression/html/._Regression-bs044.html
@@ -409,7 +409,7 @@ with \( \boldsymbol{U}\in {\mathbb{R}}^{n\times n} \), \( \boldsymbol{\Sigma}\in
 and \( \boldsymbol{V}\in {\mathbb{R}}^{p\times p} \).
 
 <p>
-We have \( \boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{I} \) and \( \boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I} \).
+The matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are unitary/orthonormal matrices, that is in case the matrices are real we have \( \boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{U}\boldsymbol{U}^T=\boldsymbol{I} \) and \( \boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I} \).
 
 <p>
 <p>
diff --git a/doc/pub/Regression/html/Regression-reveal.html b/doc/pub/Regression/html/Regression-reveal.html
index fb956bca5..8d6fb0c6d 100644
--- a/doc/pub/Regression/html/Regression-reveal.html
+++ b/doc/pub/Regression/html/Regression-reveal.html
@@ -2035,9 +2035,14 @@ Similarly, <a href="https://arxiv.org/abs/1803.08823" target="_blank">Mehta et a
 <span style="color: #CD5555">    numpy and scipy.linalg at the cost of being slower.</span>
 <span style="color: #CD5555">    &#39;&#39;&#39;</span>
     U, s, VT = np.linalg.svd(A)
+<span style="color: #228B22">#    print(&#39;test U&#39;)</span>
+<span style="color: #228B22">#    print( (np.transpose(U) @ U - U @np.transpose(U)))</span>
+<span style="color: #228B22">#    print(&#39;test VT&#39;)</span>
+<span style="color: #228B22">#    print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))</span>
     <span style="color: #8B008B; font-weight: bold">print</span>(U)
     <span style="color: #8B008B; font-weight: bold">print</span>(s)
     <span style="color: #8B008B; font-weight: bold">print</span>(VT)
+
     D = np.zeros((<span style="color: #658b00">len</span>(U),<span style="color: #658b00">len</span>(VT)))
     <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(<span style="color: #B452CD">0</span>,<span style="color: #658b00">len</span>(VT)):
         D[i,i]=s[i]
@@ -2104,7 +2109,7 @@ with \( \boldsymbol{U}\in {\mathbb{R}}^{n\times n} \), \( \boldsymbol{\Sigma}\in
 and \( \boldsymbol{V}\in {\mathbb{R}}^{p\times p} \).
 
 <p>
-We have \( \boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{I} \) and \( \boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I} \).
+The matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are unitary/orthonormal matrices, that is in case the matrices are real we have \( \boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{U}\boldsymbol{U}^T=\boldsymbol{I} \) and \( \boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I} \).
 </section>
 
 
diff --git a/doc/pub/Regression/html/Regression-solarized.html b/doc/pub/Regression/html/Regression-solarized.html
index 5f4d11cdd..76339995a 100644
--- a/doc/pub/Regression/html/Regression-solarized.html
+++ b/doc/pub/Regression/html/Regression-solarized.html
@@ -2039,9 +2039,14 @@ Similarly, <a href="https://arxiv.org/abs/1803.08823" target="_blank">Mehta et a
 <span style="color: #CD5555">    numpy and scipy.linalg at the cost of being slower.</span>
 <span style="color: #CD5555">    &#39;&#39;&#39;</span>
     U, s, VT = np.linalg.svd(A)
+<span style="color: #228B22">#    print(&#39;test U&#39;)</span>
+<span style="color: #228B22">#    print( (np.transpose(U) @ U - U @np.transpose(U)))</span>
+<span style="color: #228B22">#    print(&#39;test VT&#39;)</span>
+<span style="color: #228B22">#    print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))</span>
     <span style="color: #8B008B; font-weight: bold">print</span>(U)
     <span style="color: #8B008B; font-weight: bold">print</span>(s)
     <span style="color: #8B008B; font-weight: bold">print</span>(VT)
+
     D = np.zeros((<span style="color: #658b00">len</span>(U),<span style="color: #658b00">len</span>(VT)))
     <span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(<span style="color: #B452CD">0</span>,<span style="color: #658b00">len</span>(VT)):
         D[i,i]=s[i]
@@ -2106,7 +2111,7 @@ with \( \boldsymbol{U}\in {\mathbb{R}}^{n\times n} \), \( \boldsymbol{\Sigma}\in
 and \( \boldsymbol{V}\in {\mathbb{R}}^{p\times p} \).
 
 <p>
-We have \( \boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{I} \) and \( \boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I} \).
+The matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are unitary/orthonormal matrices, that is in case the matrices are real we have \( \boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{U}\boldsymbol{U}^T=\boldsymbol{I} \) and \( \boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I} \).
 
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
diff --git a/doc/pub/Regression/html/Regression.html b/doc/pub/Regression/html/Regression.html
index d34f9242f..636ef6656 100644
--- a/doc/pub/Regression/html/Regression.html
+++ b/doc/pub/Regression/html/Regression.html
@@ -2044,9 +2044,14 @@ Similarly, <a href="https://arxiv.org/abs/1803.08823" target="_blank">Mehta et a
 <span style="color: #BA2121; font-style: italic">    numpy and scipy.linalg at the cost of being slower.</span>
 <span style="color: #BA2121; font-style: italic">    &#39;&#39;&#39;</span>
     U, s, VT <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>svd(A)
+<span style="color: #408080; font-style: italic">#    print(&#39;test U&#39;)</span>
+<span style="color: #408080; font-style: italic">#    print( (np.transpose(U) @ U - U @np.transpose(U)))</span>
+<span style="color: #408080; font-style: italic">#    print(&#39;test VT&#39;)</span>
+<span style="color: #408080; font-style: italic">#    print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))</span>
     <span style="color: #008000; font-weight: bold">print</span>(U)
     <span style="color: #008000; font-weight: bold">print</span>(s)
     <span style="color: #008000; font-weight: bold">print</span>(VT)
+
     D <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(U),<span style="color: #008000">len</span>(VT)))
     <span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">0</span>,<span style="color: #008000">len</span>(VT)):
         D[i,i]<span style="color: #666666">=</span>s[i]
@@ -2111,7 +2116,7 @@ with \( \boldsymbol{U}\in {\mathbb{R}}^{n\times n} \), \( \boldsymbol{\Sigma}\in
 and \( \boldsymbol{V}\in {\mathbb{R}}^{p\times p} \).
 
 <p>
-We have \( \boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{I} \) and \( \boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I} \).
+The matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are unitary/orthonormal matrices, that is in case the matrices are real we have \( \boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{U}\boldsymbol{U}^T=\boldsymbol{I} \) and \( \boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I} \).
 
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
diff --git a/doc/pub/Regression/ipynb/Regression.ipynb b/doc/pub/Regression/ipynb/Regression.ipynb
index 248e57928..ca9ad0e44 100644
--- a/doc/pub/Regression/ipynb/Regression.ipynb
+++ b/doc/pub/Regression/ipynb/Regression.ipynb
@@ -2676,9 +2676,14 @@
     "    numpy and scipy.linalg at the cost of being slower.\n",
     "    '''\n",
     "    U, s, VT = np.linalg.svd(A)\n",
+    "#    print('test U')\n",
+    "#    print( (np.transpose(U) @ U - U @np.transpose(U)))\n",
+    "#    print('test VT')\n",
+    "#    print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))\n",
     "    print(U)\n",
     "    print(s)\n",
     "    print(VT)\n",
+    "\n",
     "    D = np.zeros((len(U),len(VT)))\n",
     "    for i in range(0,len(VT)):\n",
     "        D[i,i]=s[i]\n",
@@ -2748,7 +2753,7 @@
     "with $\\boldsymbol{U}\\in {\\mathbb{R}}^{n\\times n}$, $\\boldsymbol{\\Sigma}\\in {\\mathbb{R}}^{n\\times p}$\n",
     "and $\\boldsymbol{V}\\in {\\mathbb{R}}^{p\\times p}$.\n",
     "\n",
-    "We have $\\boldsymbol{U}^T\\boldsymbol{U}=\\boldsymbol{I}$ and $\\boldsymbol{V}^T\\boldsymbol{V}=\\boldsymbol{V}\\boldsymbol{V}^T=\\boldsymbol{I}$.\n",
+    "The matrices $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are unitary/orthonormal matrices, that is in case the matrices are real we have $\\boldsymbol{U}^T\\boldsymbol{U}=\\boldsymbol{U}\\boldsymbol{U}^T=\\boldsymbol{I}$ and $\\boldsymbol{V}^T\\boldsymbol{V}=\\boldsymbol{V}\\boldsymbol{V}^T=\\boldsymbol{I}$.\n",
     "\n",
     "## Spectral Decomposition of the OLS\n",
     "\n",
diff --git a/doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz b/doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz
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diff --git a/doc/pub/Regression/pdf/Regression-minted.pdf b/doc/pub/Regression/pdf/Regression-minted.pdf
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diff --git a/doc/src/Regression/Regression.do.txt b/doc/src/Regression/Regression.do.txt
index f7dd7f5e5..646c4abf9 100644
--- a/doc/src/Regression/Regression.do.txt
+++ b/doc/src/Regression/Regression.do.txt
@@ -1590,9 +1590,14 @@ def SVDinv(A):
     numpy and scipy.linalg at the cost of being slower.
     '''
     U, s, VT = np.linalg.svd(A)
+#    print('test U')
+#    print( (np.transpose(U) @ U - U @np.transpose(U)))
+#    print('test VT')
+#    print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))
     print(U)
     print(s)
     print(VT)
+
     D = np.zeros((len(U),len(VT)))
     for i in range(0,len(VT)):
         D[i,i]=s[i]
@@ -1652,7 +1657,7 @@ We have our design matrix
 with $\bm{U}\in {\mathbb{R}}^{n\times n}$, $\bm{\Sigma}\in {\mathbb{R}}^{n\times p}$
 and $\bm{V}\in {\mathbb{R}}^{p\times p}$.
 
-We have $\bm{U}^T\bm{U}=\bm{I}$ and $\bm{V}^T\bm{V}=\bm{V}\bm{V}^T=\bm{I}$.
+The matrices $\bm{U}$ and $\bm{V}$ are unitary/orthonormal matrices, that is in case the matrices are real we have $\bm{U}^T\bm{U}=\bm{U}\bm{U}^T=\bm{I}$ and $\bm{V}^T\bm{V}=\bm{V}\bm{V}^T=\bm{I}$.
 
 !split
 ===== Spectral Decomposition of the OLS =====