Thank you, but I want to get the solutions that express a _ i (i=0,1,2,3) as the functions of elements of X _ ij, e.g.:

a_0 = (X_11 + X22 ) /2
a_1 = (X_12 + X21 ) /2
a_2 = (X_21 + X12 ) / (2 I)
a_3 = (X_11 - X22 ) /2

Regards, Zhao

By running your code, I obtained the following results:

In[50]:= 
X = {{Subscript[a, 0] + Subscript[a, 3], 
    Subscript[a, 1] - I Subscript[a, 2]}, {Subscript[a, 1] + 
     I Subscript[a, 2], Subscript[a, 0] - Subscript[a, 3]}};
coeffs = {Subscript[a, 0], Subscript[a, 1], Subscript[a, 2], 
   Subscript[a, 3]};
matrixA = Array[b, {4, 4}];
matrixXgeneric = Array[x, {2, 2}];
solA = Solve[
  Eliminate[{matrixA . (X // Flatten) == coeffs, X == matrixXgeneric},
    coeffs], Flatten[matrixA]]
matrixSol = matrixA /. solA[[1]] /. b[__] :> 1
matrixSol . (matrixXgeneric // Flatten) == coeffs /. 
  Thread[Flatten[matrixXgeneric] -> Flatten[X]] // Simplify


During evaluation of In[50]:= Solve::svars: Equations may not give solutions for all "solve" variables.

Out[54]= {{b[1, 4] -> (
   x[1, 1] - 2 b[1, 1] x[1, 1] - 2 b[1, 2] x[1, 2] - 
    2 b[1, 3] x[2, 1] + x[2, 2])/(2 x[2, 2]), 
  b[2, 4] -> (-2 b[2, 1] x[1, 1] + x[1, 2] - 2 b[2, 2] x[1, 2] + 
    x[2, 1] - 2 b[2, 3] x[2, 1])/(2 x[2, 2]), 
  b[3, 4] -> (-2 b[3, 1] x[1, 1] + I x[1, 2] - 2 b[3, 2] x[1, 2] - 
    I x[2, 1] - 2 b[3, 3] x[2, 1])/(2 x[2, 2]), 
  b[4, 4] -> (
   x[1, 1] - 2 b[4, 1] x[1, 1] - 2 b[4, 2] x[1, 2] - 
    2 b[4, 3] x[2, 1] - x[2, 2])/(2 x[2, 2])}}

Out[55]= {{1, 1, 1, (-x[1, 1] - 2 x[1, 2] - 2 x[2, 1] + x[2, 2])/(
  2 x[2, 2])}, {1, 1, 1, (-2 x[1, 1] - x[1, 2] - x[2, 1])/(
  2 x[2, 2])}, {1, 1, 
  1, (-2 x[1, 1] - (2 - I) x[1, 2] - (2 + I) x[2, 1])/(
  2 x[2, 2])}, {1, 1, 
  1, (-x[1, 1] - 2 x[1, 2] - 2 x[2, 1] - x[2, 2])/(2 x[2, 2])}}

Out[56]= True

But I still cannot see how a _ i (i=0,1,2,3) are expressed via X _ ij by your results. But I still can't see how your result represents a _ i (i=0,1,2,3) by X _ ij (i=1,2, j = 1,2).

Regards, Zhao

Wonderful, thank you very much for your tip. It does match up perfectly with the results I provided earlier:

In[11]:= 
matrixSol . (matrixXgeneric // Flatten) == coeffs // Simplify

Out[11]= {1/2 (x[1, 1] + x[2, 2]), 1/2 (x[1, 2] + x[2, 1]), 
  1/2 I (x[1, 2] - x[2, 1]), 1/2 (x[1, 1] - x[2, 2])} == {Subscript[a,
   0], Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}

The above result is the same as my previous representation. Is this just a coincidence?

Regards, Zhao

Wonderful confirmation and validation. This further prompts me to consider whether the issue here is related to the following discussion:

Regards, Zhao