Gradient of quadratic vector function

Question

I'm struggling with this very simple gradient question.

Where...

Now I've already been told the answer is Ax + b, but I can't work out how to get to the first term. Intuitively it looks like a quadratic equation, but when I go through the expansion of 1/2 xAx, for each element, then differentiate, I get left with:

Or in other words AIx + Ax

What am I missing?

Answer 1

The top row of your derivative is incorrect. For example, the coefficient of the $x_2$ term for example will be $(A_{12} + A_{21})$ not just $A_{12}$. This is because the expression $x^TAx$ contains $A_{12}x_1x_2 + A_{21}x_2x_1$. You considered the first term but neglected the second.