Assuming that A is a real symmetric n x n matrix, and
prove that f has a critical point at x if and only if Ax = ?.
I need to find out what Ax must equal for a critical point, and I just can't come up with anything.
I've tried taking the gradient and such, but I always end up with
I feel like that's not the right answer, though.
I don't understand how I get the gradient now. Doesn't setting x as that make it a scalar? Doesn't it need to be a vector? Can you please help me? I really have no clue.
But how then, do I know what to do with A when I take the gradient? I learned that that gradient of is 2Ax. But then I can't figure out what Ax must be in order for there to be a critical point. I don't know how Writing the vector in component form changes anything either. I'll keep trying though.
I don't get what is. Can you please explain where that comes from? I also don't get what I'd do to take the gradient of that equation. Can you please explain what I need to do to take the gradient? I guess part of my problem is I don't understand what the A component of the sum is.
Thanks! Sorry for all the lack of understanding. I'm doing my best.
At this point I got it down to
But I don't know how to simplifiy it past that point. This is where I had gotten it to when not in component form I think. But now I just don't know where to go.
I'm so confused. Thank you for all your help so far. It's so greatly appreciated.