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How to reduce the number of independent variables in mathematics

I don’t agree if this is really a mathematical question or actually a mathematical question .: D

suppose i have a matrix

{{4/13 + (9 w11)/13 + (6 w12)/13, 
  6/13 + (9 w21)/13 + (6 w22)/13}, {-(6/13) + (6 w11)/13 + (4 w12)/
   13, -(9/13) + (6 w21)/13 + (4 w22)/13}}

with w11, w12, w21, w22as free parameters.

And from visual control, I know that it 3*w11+2*w12can be represented as one variable, but 3*w21+2*w22can be represented as another. Thus, essentially this matrix has only two independent variables. For any matrix of this form, is there a way to automatically reduce the number of independent variables? I think I was fixated on formulating it exactly in a mathematical way.

Share your thoughts. Many thanks.

Edit:

My question is really the following. Such a matrix

{{4/13 + (9 w11)/13 + (6 w12)/13, 
  6/13 + (9 w21)/13 + (6 w22)/13}, {-(6/13) + (6 w11)/13 + (4 w12)/
   13, -(9/13) + (6 w21)/13 + (4 w22)/13}}


{{a+4/13 + (9 w11)/13 + (6 w12)/13, 
  6/13*c + (9 w21)/13 + (6 w22)/13}, {-(6/13)/d + (6 w11)/13 + (4 w12)/
   13, -(9/13) + (6 w21)/13 + (4 w22)/13}}

n ( 2), y1, y2,..., yn, y1, y2,..., yn w11, w12, w21, w22.

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1

mat = {{4/13 + (9 w11)/13 + (6 w12)/13,6/13 + (9 w21)/13 + (6 w22)/13},
  {-(6/13) + (6 w11)/13 + (4 w12)/13, -(9/13) + (6 w21)/13 + (4 w22)/13}};

, , .

mat2 = Array[y, Dimensions[mat]];

( ) , mat-mat2 == 0. . Eliminate; GroebnerBasis.

GroebnerBasis[Flatten[mat - mat2], Variables[mat2], Variables[mat]]

Out [59] = {-3 + 2 y [1, 2] - 3 y [2, 2], -2 + 2 y [1, 1] - 3 y [2, 1]}

, .

--- ---

, . , .

gb = GroebnerBasis[Flatten[mat - mat2], Variables[mat2], Variables[mat]];
vars = Flatten[mat2];

PolynomialReduce[vars, gb, vars][[All, 2]]

Out [278] = {1 + 3/2 y [2, 1], 3/2 + 3/2 y [2, 2], y [2, 1], y [2, 2]}

--- ---

Wolfram Research

+10

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