I have a question about minimizing a metric, i have trouble to find in the back of my head which principle i should apply or how to tackle this problem really.
I have a vector $X$ and a vector $Y$ of lengths $n$, consisting of complex numbers. I now want to find a constant $a$ (i guess also this will be complex?) that minimizes the function $\sum^{n}_{1}|X-aY|^2$ (e.q 1).
The problem this derives from is that i need to minimize $$\sum_{x_1}\sum_{x_2}\sqrt{|E_{reconstructed}\left(x_1,x_2\right)-E_s\left(x_1\right)E_{ref}\left(x_1,x_2\right)|^2}$$ with regards to $E_s\left(x_1\right)$. My approach so far is to solve each position for $E_s\left(x_1\right)$ by solving the problem $$|E_{reconstructed}\left(x_1,(1...n)\right)-E_s\left(x_1\right)E_{ref}\left(x_1,(1...n)\right)|$$, i.e iteratively for each index $x_1$ with the vectors containing all of the values of $x_2$. To clarify, i know $E_{reconstructed}\left(x_1,x_2\right)$ and $E_{ref}\left(x_1,x_2\right)$ and need to solve for the $E_s\left(x_1\right)$ that minimizes (e.q 1) in the case where x and y are replaced with the E-functions given just above here.
Thanks in advance for any help you might offer me.
This is not a hw quesiton.
EDIT Am i on the right track if i'm thinking ... least squares ?