Multidimensional Matrix Newton's Method problem

Question

I included 3 pictures (both as google drive or image below):

1. Implementing Newton's method.

• 2

2. Derivative\Jacobian of F.

• 3

3. The Assignment.

• 4

My question for (1) \ (2) is: Why are we able to have a 2x2 matrix for Jacobian? How does the derivative of $$\frac{1}{|| x - a||}$$ yield: $$(x-a)^T$$ (see (2) ) in particular on the right side to give an outer product?

• I understand why these are necessary dimensionally, just not sure where they come from. We were given the F and F' so I really want to understand the derivation.

For (1) does my algorithm correctly implement one iteration? I set my own constants.

• I'm used to Newton for single and for multidimensional and multiple equations. But in this case, it appears there is only one F equation, with multiple dimensions. yet somehow we 'create' a second equation when doing the Jacobian.
• Also in practice, I don't plan on using the inverse Jacobian I would bring it to the other side, but for this pseudo-code, it shouldn't matter.

Main question: What are the 2 F equations? They seem to yield the 2x2 Jacobian. Derivatives with column vectors?

Thanks in advance