Custom categorical cross entropy

#1

I want to define custom cross entropy loss penalizing different class errors.

Categorical cross entropy loss =CodeCogsEqn%20(2)

I want to give different weights to different prediction errors. e.g. For a 3-class problem, below cost matrix is defined.
CodeCogsEqn%20(4)

Is it fair to define new cost entropy loss function as below?
CodeCogsEqn%20(5)

Where C_{ij} - Cost of classifying class ‘i’ as class ‘j’

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