title: Scikit-Learn PCA
body: <p>I am using input data from <a href="http://www.cs.otago.ac.nz/cosc453/student_tutorials/principal_components.pdf">here</a> (see Section 3.1).</p>
  
  <p>I am trying to reproduce their covariance matrix, eigenvalues, and eigenvectors using scikit-learn. However, I am unable to reproduce the results as presented in the data source. I've also seen this input data elsewhere but I can't discern whether it's a problem with scikit-learn, my steps, or the data source.</p>
  
  <pre><code>data = np.array([[2.5,2.4],
                   [0.5,0.7],
                   [2.2,2.9],
                   [1.9,2.2],
                   [3.1,3.0],
                   [2.3,2.7],
                   [2.0,1.6],
                   [1.0,1.1],
                   [1.5,1.6],
                   [1.1,0.9],
                   ]) 
  
  centered_data = data-data.mean(axis=0)
  pca = PCA()
  pca.fit(centered_data)
  print(pca.get_covariance()) #Covariance Matrix
  
  array([[ 0.5549,  0.5539],
     [ 0.5539,  0.6449]])
  
  print(pca.explained_variance_ratio_) #Eigenvalues (normalized)
  
  [ 0.96318131  0.03681869]
  
  print(pca.components_) #Eigenvectors
  
  [[-0.6778734  -0.73517866]
   [ 0.73517866 -0.6778734 ]]
  </code></pre>
  
  <p>Surprisingly, the projections matches the results from the data source described above.</p>
  
  <pre><code>print(pca.transform(centered_data)) #Projections
  
  array([[-0.82797019,  0.17511531],
     [ 1.77758033, -0.14285723],
     [-0.99219749, -0.38437499],
     [-0.27421042, -0.13041721],
     [-1.67580142,  0.20949846],
     [-0.9129491 , -0.17528244],
     [ 0.09910944,  0.3498247 ],
     [ 1.14457216, -0.04641726],
     [ 0.43804614, -0.01776463],
     [ 1.22382056,  0.16267529]])
  </code></pre>
  
  <p>Here is what I don't understand:</p>
  
  <ol>
  <li>Why is the covariance matrix is different?</li>
  <li><strong>Updated</strong>: How do I get eigenvalues from scikit-learn that are not already normalized? </li>
  </ol>