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I'm struggling with this very simple gradient question.

enter image description here

Where...

enter image description here

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:

enter image description here

Or in other words AIx + Ax

What am I missing?

Rodrigo de Azevedo
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Will Taylor
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  • I would expect a $2$ in front of every coefficient as the derivative of $x^2$ is $2x$ and this is a quadradic form. I expect your error lays in taking the derivative. – CyclotomicField Dec 24 '21 at 12:48
  • Well, taking the first term which before differentiation is: x_1^2 A11 + x_1*x_2 A12... etc. Then when we partially differentiate this by x_1 we get the first line in the above snapshot, no? – Will Taylor Dec 24 '21 at 12:59
  • Did you use that the matrix is symmetric? This simplifies some terms. – CyclotomicField Dec 24 '21 at 13:04
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    Excuse me, found my mistake... Thanks for the help. :) – Will Taylor Dec 24 '21 at 13:10

1 Answers1

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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.

Rioghasarig
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