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I have some problems.

First of all, I declare matrices like below

$Assumptions = {
  Element[X, Matrices[{m, n}]], 
  Element[T, Matrices[{n, k}]], 
  Element[Y, Matrices[{m, k}]]};

and I want to apply

transpose(X T - Y).(X T - Y) = 
  transpose[T].transpose[X].X.T - transpose[T].transpose[X].Y - 
  transpose[Y].X.T + transpose[Y].Y

so I tried

  1. Inner[Times, {X \[Transpose]}, {T \[Transpose]}, Plus] - \[Transpose][Y]

  2. Inner[NonCommutativeMultiply, {X}, {T}] - {Y} = {-Y + X ** T} ...

But I can not multiply the two above expressions.

Inner[NonCommutativeMultiply, 
  Inner[Times, {X\[Transpose]}, {T\[Transpose]}, Plus] - Transpose[Y], 
  Inner[NonCommutativeMultiply, {X}, {T}] - {Y}]

because I get errors. Could you help me?

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Works perfectly well for me. The only thing I see that is wrong is that you forgot the braces for Transpose[Y], so the code should look like this:

Inner[NonCommutativeMultiply, 
Inner[Times, {X\[Transpose]}, {T\[Transpose]}, Plus] - {Transpose[Y]}, 
Inner[NonCommutativeMultiply, {X}, {T}] - {Y}]
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