some update of pca

doc/src/DimRed/DimRed.do.txt

@@ -812,7 +812,7 @@ and the mean-centered data $\bar{X} = \{ \bar{x}_1, \ldots, \bar{x}_n \}$ takes
 When you are done with these steps, print out $\mu_n$ to verify it is
 close to $\mu$ and plot your mean centered data to verify it is
 centered at the origin! Compare your code with the functionality from _Scikit-Learn_ discussed above.
-The following code elements perform these operations using _pandas_ or your own functionality for doing so. 
+The following code elements perform these operations using _pandas_ or using our own functionality for doing so. The latter, using _numpy_ is rather simply through the _mean()_ function. 
 !bc pycod
 df = pd.DataFrame(X)
 # Pandas does the centering for us
@@ -828,7 +828,7 @@ Alternatively, you could also have used the functions we discussed earlier for s
 That is, we could have used the _StandardScaler_ function in _Scikit-Learn_, a function which 
 ensures that for each feature/predictor we study the mean value is
 zero and the variance is one (every column in the design/feature
-matrix).
+matrix).  You would then not get the same results, since we divide by the variance. For diagonal covariance matrix elements will then be one, while the non-diagonal ones will be divided by $2\sqrt{2}$ for our specific case.
 
 === Compute the sample covariance ===
 
@@ -860,12 +860,13 @@ following tasks:
 * We plot the mean centered data and lines along the first and second principal components
 * Then we project the mean centered data onto the first and second principal components, and plot the projected data. 
 * Finally, we approximate the data as
+
 !bt
 \begin{equation*}
-x_i \approx \tilde{x}_i := \mu_n + \langle x_i, v_0 \rangle v_0
+x_i \approx \tilde{x}_i = \mu_n + \langle x_i, v_0 \rangle v_0
 \end{equation*}
 !et
-where $v_0$ is the first principal component. What do you observe?
+where $v_0$ is the first principal component. 
 
 Collecting all these steps we can write our own PCA function and
 compare this with the functionality included in _Scikit-Learn_.