Matrix Differentiation proof

Question

I got stuck. Can someone please tell me how equation 46 was gotten from equation 45?

Answer 1

$\alpha=\sum_{j=1}^{n}\sum_{i=1}^{n}a_{ij}x_ix_j$

For $j=k$, the part of summation, for which the derivative is not zero, is

$$\sum_{i=1}^{n}a_{ik}x_ix_k$$

The derivative is

$$\sum_{i=1,i\neq k}^{n}a_{ik}x_i+2a_{kk}x_k$$

For $j\neq k$ and $i=k$, the part of summation, for which the derivative is not zero, is

$$\sum_{j=1, j \neq k}^{n}a_{kj}x_kx_j$$

The derivative is

$$\sum_{j=1, j \neq k}^{n}a_{kj}x_j$$

In total, you have

$$\frac{\delta \alpha}{\delta x_k}=\sum_{i=1,i\neq k}^{n}a_{ik}x_i+2a_{kk}x_k+\sum_{j=1, j \neq k}^{n}a_{kj}x_j$$

$$\frac{\delta \alpha}{\delta x_k}=(\sum_{i=1,i\neq k}^{n}a_{ik}x_i+a_{kk}x_k)+(\sum_{j=1, j \neq k}^{n}a_{kj}x_j+a_{kk}x_k)$$

$$\frac{\delta \alpha}{\delta x_k}=(\sum_{i=1}^{n}a_{ik}x_i)+(\sum_{j=1}^{n}a_{kj}x_j)$$