Update week35.do.txt

doc/src/week35/week35.do.txt

@@ -570,7 +570,9 @@ From this we have, using the definition of the Jacobian
 !split
 ===== Example 2 =====
 
-We define a scalar (our cast functions are in general also scalars, think of the mean squared error) as the result of some matrix vector multiplications
+We define a scalar (our cost/loss functions are in general also scalars,
+just think of the mean squared error) as the result of some matrix vector
+multiplications
 
 !bt
 \[
@@ -601,6 +603,55 @@ Since $\alpha$ is a scalar we have $\alpha =\alpha^T=\bm{x}^T\bm{A}^T\bm{y}$. De
 
 
 
+!split
+===== Example 3 =====
+
+We start with a new scalar but where now the vector $\bm{y}$ is
+replaced by a vector $\bm{x}$ and the matrix $\bm{A}$ is a square
+matrix with dimension $n\times n$.
+
+!bt
+\[
+\alpha = \bm{x}^T\bm{A}\bm{x}$,
+\]
+!et
+with $\bm{x}$ a vector of length $n$.
+
+We write out the specific sums involved in the calculation of $\alpha$
+!bt
+\[
+\alpha = \sum_{i=0}^{n-1}\sum_{j=0}^({n-1}x_i a_{ij}x_j, 
+\]
+!et
+taking the derivative of $\alpha$ with respect to a given component $x_k$ we get the two sums
+!bt
+\[
+\frac{\partial \alpha}{\partial x_k}  = \sum_{i=0}^{n-1}a_{ik}x_i+\sum_{j=0}^{n-1}a_{kj}x_j, 
+\]
+!et
+for $\all k =0,1,2,\dots,n-1$. We identify these sums as
+!bt
+\[
+\frac{\partial \alpha}{\partial \bm{x}}  = \bm{x}^T\left(\bm{A}^T+\bm{A}\right).
+\]
+!et
+If the matrix $\bm{A}$ is symmetric, that is $\bm{A}=\bm{A}^T$, we have 
+!bt
+\[
+\frac{\partial \alpha}{\partial \bm{x}}  = 2\bm{x}^T\bm{A}.
+\]
+!et
+
+!split
+===== Example 4 =====
+
+We let the scalar $\alpha$ be defined by
+!bt
+\[
+\alpha = \bm{y}^T\bm{x},
+\]
+!et
+where both $\bm{y}$ and $\bm{x}$ have the same length $n$, or if we wish to think of them as column vectors, they have dimensions $n\times 1$
 
 !bt
 \[