more typos

doc/src/week35/week35.do.txt

@@ -1695,13 +1695,13 @@ and using the orthogonality of the matrix $\bm{U}$ we have
 \bm{X}^T\bm{X}=\bm{V}\bm{\Sigma}^T\bm{\Sigma}\bm{V}^T.
 \]
 !et
-We define $\bm{\Sigma}^T\bm{\sigma}=\bm{D}^2, where $\bm{D}$ is a diagonal matrix containing only the singular values squared. It has dimensionality $p \times p$.
+We define $\bm{\Sigma}^T\bm{\Sigma}=\tilde{\bm{\Sigma}}^2$ which is  a diagonal matrix containing only the singular values squared. It has dimensionality $p \times p$.
 
 This means, using the orthogonality of $\bm{V}$,  that we get
 
 !bt
 \[
-\bm{X}^T\bm{X}=\bm{D}^2.
+\bm{X}^T\bm{X}=\tilde{\bm{\Sigma}}^2.
 \]
 !et
 
@@ -1716,7 +1716,7 @@ and using our SVD decomposition of $\bm{X}$ we have
 
 !bt
 \[
-\tilde{y}_{\mathrm{OLS}}=\bm{U}\bm{\Sigma}\bm{V}^T\bm{D}^{-2}\bm{V}\bm{\Sigma}^T\bm{U}^T\bm{y},
+\tilde{y}_{\mathrm{OLS}}=\bm{U}\bm{\Sigma}\bm{V}^T\tilde{\bm{\Sigma}}^{-2}\bm{V}\bm{\Sigma}^T\bm{U}^T\bm{y},
 \]
 !et
 which gives us, using the orthogonality of the matrices $\bm{U}$ and $\bm{V}$,
@@ -1733,7 +1733,7 @@ Note here that when we perform the  multiplication of the various matrices, the
 \bm{U}=[\bm{u}_0,\bm{u}_1,\dots,\bm{u}_{n-1}],
 \]
 !et
-that belong to $i>p-1$, give all zeros when we perform the multiplications. This means the sum above has non-zero elements only up to $i=p-1$. This corresponds also to the number of singular values (these are all non-zero).
+that belong to $i>p-1$, result in only  zeros when we perform the multiplications. This means that the sum above has non-zero elements only up to $i=p-1$. This corresponds also to the number of singular values (these are all non-zero).
 
 It means that the ordinary least square model (with the optimal parameters) $\bm{\tilde{y}}$, corresponds to an orthogonal transformation of the output (or target) vector $\bm{y}$ by the vectors of the matrix $\bm{U}$.