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Suppose I have an nxn matrix A. (If needed it can be assumed invertible). I can perform a transform on the matrix in the following way:

D=C*A*C^-1. C can be chosen to be any nxn invertible matrix.

Does this transform have any meaning, which can be easily understood or visualized?

What space is covered by possible values of D for a given A?

What is the minimal set of matrices A, parametrized by as few as possible parameters, which covers all possible matrices D?

2x2 case is of particular interest, but a general answer would certainly be useful.

Thanks