Finding linear mapping between two matrices

Questions
1

Hello,
I recently started to use Mathematica and I wonder if we have a method to find linear mapping between two matrices in Mathematica.
For example:
If I have those 2 matrices:

idealrho.png

xrho.png

Then, I will have this linear mapping as matrix:

map.png
map.png130×100 1.1 KB

This is the simplest case. I have matrices with different coefficients and higher dimensions like 16*16.. This is why I wonder if we have a method in Mathematica which can do that.. Or how can I do this map in Mathematica?

Thanks in advance.
[1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=idealrho.png&userId=2481260
[2]: https://community.wolfram.com//c/portal/getImageAttachment?filename=xrho.png&userId=2481260
[3]: https://community.wolfram.com//c/portal/getImageAttachment?filename=map.png&userId=2481260

2

For the example given, there are four equations, and 16 variables. So you will need to constrain the problem more if you want a unique solution.

If you treated it as a transformation using a 2 x 2 matrix, it could be done like so.

amat = Array[a, {2, 2}];
romat = Array[rho, {2, 2}, 0];
Solve[amat . 
   romat == {{rho[1, 1], rho[1, 0]}, {rho[0, 1], rho[0, 0]}}, 
 Flatten[amat]]

(* Out[11]= {{a[1, 1] -> -((-rho[1, 0]^2 + rho[1, 1]^2)/(
    rho[0, 1] rho[1, 0] - rho[0, 0] rho[1, 1])), 
  a[1, 2] -> -((rho[0, 0] rho[1, 0] - rho[0, 1] rho[1, 1])/(
    rho[0, 1] rho[1, 0] - rho[0, 0] rho[1, 1])), 
  a[2, 1] -> -((-rho[0, 0] rho[1, 0] + rho[0, 1] rho[1, 1])/(
    rho[0, 1] rho[1, 0] - rho[0, 0] rho[1, 1])), 
  a[2, 2] -> -((rho[0, 0]^2 - rho[0, 1]^2)/(
    rho[0, 1] rho[1, 0] - rho[0, 0] rho[1, 1]))}}*)
3

Thanks!