week43 update

doc/src/week43/week43.do.txt

@@ -2720,7 +2720,7 @@ matrix without them.
 We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix $\bm{X}$ as 
 !bt
 \[
-\bm{C}[\bm{x}] = \frac{1}{n}\bm{X}\bm{X}^T= \mathbb{E}[\bm{X}\bm{X}^T].
+\bm{C}[\bm{x}] = \frac{1}{n}\bm{X}^T\bm{X}= \mathbb{E}[\bm{X}^T\bm{X}].
 \]
 !et
 
@@ -2739,7 +2739,7 @@ x_{10} & x_{11}\\
 If we then compute the expectation value
 !bt
 \[
-\mathbb{E}[\bm{X}\bm{X}^T] = \frac{1}{n}\bm{X}\bm{X}^T=\begin{bmatrix}
+\mathbb{E}[\bm{X}^T\bm{X}] = \frac{1}{n}\bm{X}^T\bm{X}=\begin{bmatrix}
 x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\
 x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\
 \end{bmatrix},
@@ -2764,31 +2764,31 @@ It is easy to generalize this to a matrix $\bm{X}\in {\mathbb{R}}^{n\times p}$.
 We have that the covariance matrix (the correlation matrix involves a simple rescaling) is given as
 !bt
 \[
-\bm{C}[\bm{x}] = \frac{1}{n}\bm{X}\bm{X}^T= \mathbb{E}[\bm{X}\bm{X}^T].
+\bm{C}[\bm{x}] = \frac{1}{n}\bm{X}^T\bm{X}= \mathbb{E}[\bm{X}^T\bm{X}].
 \]
 !et
 Let us now assume that we can perform a series of orthogonal transformations where we employ some orthogonal matrices $\bm{S}$.
 These matrices are defined as $\bm{S}\in {\mathbb{R}}^{p\times p}$ and obey the orthogonality requirements $\bm{S}\bm{S}^T=\bm{S}^T\bm{S}=\bm{I}$. The matrix can be written out in terms of the column vectors $\bm{s}_i$ as $\bm{S}=[\bm{s}_0,\bm{s}_1,\dots,\bm{s}_{p-1}]$ and $\bm{s}_i \in {\mathbb{R}}^{p}$.
 
-Assume also that there is a transformation $\bm{S}\bm{C}[\bm{x}]\bm{S}^T=\bm{C}[\bm{y}]$ such that the new matrix $\bm{C}[\bm{y}]$ is diagonal with elements $[\lambda_0,\lambda_1,\lambda_2,\dots,\lambda_{p-1}]$.  
+Assume also that there is a transformation $\bm{S}^T\bm{C}[\bm{x}]\bm{S}=\bm{C}[\bm{y}]$ such that the new matrix $\bm{C}[\bm{y}]$ is diagonal with elements $[\lambda_0,\lambda_1,\lambda_2,\dots,\lambda_{p-1}]$.  
 
 That is we have
 !bt
 \[
-\bm{C}[\bm{y}] = \mathbb{E}[\bm{S}\bm{X}\bm{X}^T\bm{S}^T]=\bm{S}\bm{C}[\bm{x}]\bm{S}^T,
+\bm{C}[\bm{y}] = \mathbb{E}[\bm{S}^T\bm{X}^T\bm{X}T\bm{S}]=\bm{S}^T\bm{C}[\bm{x}]\bm{S},
 \]
 !et
-since the matrix $\bm{S}$ is not a data dependent matrix.   Multiplying with $\bm{S}^T$ from the left we have
+since the matrix $\bm{S}$ is not a data dependent matrix.   Multiplying with $\bm{S}$ from the left we have
 !bt
 \[
-\bm{S}^T\bm{C}[\bm{y}] = \bm{C}[\bm{x}]\bm{S}^T,
+\bm{S}\bm{C}[\bm{y}] = \bm{C}[\bm{x}]\bm{S},
 \]
 !et
 and since $\bm{C}[\bm{y}]$ is diagonal we have for a given eigenvalue $i$ of the covariance matrix that
 
 !bt
 \[
-\bm{S}^T_i\lambda_i = \bm{C}[\bm{x}]\bm{S}^T_i.
+\bm{S}_i\lambda_i = \bm{C}[\bm{x}]\bm{S}_i.
 \]
 !et