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Assuming G a 4x4, C a 4x2, and DT a 2x4 matrix, I want to vectorize the following expression, i.e., to transform it into matrix notation form to accelerate its computation. For instance, C[1,1]*DT[1,3]+C[1,2]*DT[2,3] are to be replaced with G[1,3]. However, I cannot do the task using Replace or ReplaceAll. 

c = Table[C[i, j], {i, 1, 4}, {j, 1, 2}]
dt = Table[DT[i, j], {i, 1, 2}, {j, 1, 4}]
g = Table[G[i, j], {i, 1, 4}, {j, 1, 4}]
cdt = c.dt

A=(G[1, 1]^4 + (2*G[2, 2] + G[3, 3])*
    G[1, 1]^3 + (G[2, 2]^2 + G[3, 3]*G[2, 2] - 2*G[1, 2]*G[2, 1] - 
      C[1, 1]*DT[1, 3]*G[3, 1] - C[2, 1]*DT[1, 3]*G[2, 1] - 
      C[3, 1]*DT[1, 2]*G[1, 2] - C[1, 2]*DT[2, 3]*G[3, 1] - 
      C[2, 1]*DT[1, 3]*G[3, 2] - C[2, 2]*DT[2, 3]*G[2, 1] - 
      C[3, 2]*DT[2, 2]*G[1, 2] - C[2, 2]*DT[2, 3]*G[3, 2])*
    G[1, 1]^2 + (C[1, 1]*DT[1, 3]*G[2, 1]^2 - 
      G[1, 2]*G[2, 1]*G[3, 3] - 2*G[1, 2]*G[2, 1]*G[2, 2] + 
      C[1, 2]*DT[2, 3]*G[2, 1]^2 + C[1, 1]*DT[1, 2]*G[1, 2]*G[3, 1] + 
      C[2, 1]*DT[1, 2]*G[1, 2]*G[2, 1] + 
      C[1, 1]*DT[1, 3]*G[2, 1]*G[3, 2] - 
      C[1, 1]*DT[1, 3]*G[2, 2]*G[3, 1] + 
      C[1, 2]*DT[2, 2]*G[1, 2]*G[3, 1] + 
      C[2, 1]*DT[1, 2]*G[1, 2]*G[3, 2] + 
      C[2, 1]*DT[1, 3]*G[1, 2]*G[3, 1] + 
      C[2, 2]*DT[2, 2]*G[1, 2]*G[2, 1] - 
      C[3, 1]*DT[1, 2]*G[1, 2]*G[2, 2] + 
      C[1, 2]*DT[2, 3]*G[2, 1]*G[3, 2] - 
      C[1, 2]*DT[2, 3]*G[2, 2]*G[3, 1] + 
      C[2, 2]*DT[2, 2]*G[1, 2]*G[3, 2] + 
      C[2, 2]*DT[2, 3]*G[1, 2]*G[3, 1] - 
      C[3, 2]*DT[2, 2]*G[1, 2]*G[2, 2])*G[1, 1] + 
   G[1, 2]^2*G[2, 1]^2 - C[1, 1]*DT[1, 2]*G[1, 2]*G[2, 1]^2 - 
   C[1, 2]*DT[2, 2]*G[1, 2]*G[2, 1]^2 - 
   C[2, 1]*DT[1, 2]*G[1, 2]^2*G[3, 1] + 
   C[3, 1]*DT[1, 2]*G[1, 2]^2*G[2, 1] - 
   C[2, 2]*DT[2, 2]*G[1, 2]^2*G[3, 1] + 
   C[3, 2]*DT[2, 2]*G[1, 2]^2*G[2, 1] - 
   C[1, 1]*DT[1, 2]*G[1, 2]*G[2, 1]*G[3, 2] + 
   C[1, 1]*DT[1, 2]*G[1, 2]*G[2, 2]*G[3, 1] - 
   C[1, 2]*DT[2, 2]*G[1, 2]*G[2, 1]*G[3, 2] + 
   C[1, 2]*DT[2, 2]*G[1, 2]*G[2, 2]*G[3, 1])/(G[1, 1]^3 + 
   G[2, 2]*G[1, 1]^2 - G[1, 2]*G[2, 1]*G[1, 1])


Simplify[A] //. cdt -> g

Thanks for your help.

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    $\begingroup$ Do this: Collect[A, _G] /. C[x_, 1] DT[1, y_] :> G[x, y] - C[x, 2] DT[2, y] // Simplify. But also, don't use capital letters for user-defined symbols (e.g. C is a built-in Mathematica symbol, and by using it you might run into problems since it's already defined as something). $\endgroup$
    – march
    Commented Mar 23, 2017 at 3:52
  • $\begingroup$ Would you please answer my question in a more detailed version to accept it. $\endgroup$
    – remo
    Commented Mar 23, 2017 at 5:25

1 Answer 1

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Do this:

Collect[A, _G] /. C[x_, 1] DT[1, y_] :> G[x, y] - C[x, 2] DT[2, y] // Simplify
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