Reduce and Expand solutions after Solve

Question

I solve a system of 4 equations with 4 variables z_i (see below).

I would like to change the way results are written as such for each i: y_i = AX + sum[By_(-i)] + Cz_i + u_i

where the sum comes from the other indices than i, u_i gathers all the terms not in X, y_(-i), and z_i.

for instance: y_1 = AX + By_2 + Cy_3 + Dy_4 + Ez_1 + u_1

Is it possible to do so?

Thanks.

Solve[{Subscript[\[Pi], 11] Subscript[z, 1] + 
    Subscript[\[Pi], 12] Subscript[z, 2] + 
    Subscript[\[Pi], 13] Subscript[z, 3] + 
    Subscript[\[Pi], 14] Subscript[z, 4] + Subscript[\[Pi], 10] X + 
    Subscript[\[Nu], 1] == Subscript[y, 1], 
  Subscript[\[Pi], 21] Subscript[z, 1] + 
    Subscript[\[Pi], 22] Subscript[z, 2] + 
    Subscript[\[Pi], 23] Subscript[z, 3] + 
    Subscript[\[Pi], 24] Subscript[z, 4] + Subscript[\[Pi], 20] X + 
    Subscript[\[Nu], 2] == Subscript[y, 2], 
  Subscript[\[Pi], 31] Subscript[z, 1] + 
    Subscript[\[Pi], 32] Subscript[z, 2] + 
    Subscript[\[Pi], 33] Subscript[z, 3] + 
    Subscript[\[Pi], 34] Subscript[z, 4] + Subscript[\[Pi], 30] X + 
    Subscript[\[Nu], 3] == Subscript[y, 3], 
  Subscript[\[Pi], 41] Subscript[z, 1] + 
    Subscript[\[Pi], 42] Subscript[z, 2] + 
    Subscript[\[Pi], 43] Subscript[z, 3] + 
    Subscript[\[Pi], 44] Subscript[z, 4] + Subscript[\[Pi], 40] X + 
    Subscript[\[Nu], 4] == Subscript[y, 4]}, {Subscript[z, 1], 
  Subscript[z, 2], Subscript[z, 3], Subscript[z, 4]}]

Answer 1

Long for a comment, but not a complete answer...

system = {Subscript[\[Pi], 11] Subscript[z, 1] + 
Subscript[\[Pi], 12] Subscript[z, 2] + 
Subscript[\[Pi], 13] Subscript[z, 3] + 
Subscript[\[Pi], 14] Subscript[z, 4] + Subscript[\[Pi], 10] X + 
Subscript[\[Nu], 1] == Subscript[y, 1], 
Subscript[\[Pi], 21] Subscript[z, 1] + 
Subscript[\[Pi], 22] Subscript[z, 2] + 
Subscript[\[Pi], 23] Subscript[z, 3] + 
Subscript[\[Pi], 24] Subscript[z, 4] + Subscript[\[Pi], 20] X + 
Subscript[\[Nu], 2] == Subscript[y, 2], 
Subscript[\[Pi], 31] Subscript[z, 1] + 
Subscript[\[Pi], 32] Subscript[z, 2] + 
Subscript[\[Pi], 33] Subscript[z, 3] + 
Subscript[\[Pi], 34] Subscript[z, 4] + Subscript[\[Pi], 30] X + 
Subscript[\[Nu], 3] == Subscript[y, 3], 
Subscript[\[Pi], 41] Subscript[z, 1] + 
Subscript[\[Pi], 42] Subscript[z, 2] + 
Subscript[\[Pi], 43] Subscript[z, 3] + 
Subscript[\[Pi], 44] Subscript[z, 4] + Subscript[\[Pi], 40] X + 
Subscript[\[Nu], 4] == Subscript[y, 4]}

Next we Eliminate from this e.g. Subscript[y, 1] and solve for all Subscript[z, i] except Subscript[z, 1]:

Solve[Eliminate[system, Subscript[y, 1]], {Subscript[z, 2], Subscript[z, 3], 
Subscript[z, 4]}]

Now we can substitute this in first equation of system (equation on Subscript[y, 1]):

system[[1, 1]] /. 
Solve[Eliminate[system, Subscript[y, 1]], {Subscript[z, 2], 
Subscript[z, 3], Subscript[z, 4]}][[1]]

and collect terms:

coly1=Collect[%, {X, Subscript[y, 2], Subscript[y, 3], Subscript[y, 4], 
Subscript[z, 1]}, FullSimplify]

The result is complicated enough, but in the desired form. You can use Coefficient[coly1,X] to get your A etc. The same technique can be use for Subscript[y, 2], Subscript[y, 3], Subscript[y, 4].