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I know that if T(linear transformation in finite dim-vector space) is diagonalizable, then the matrix A that represent T is diagonalizable if there exist a matrix P=! 0 that is invertible and A=P^-1 * D * P for a diagonal matrix D.

I also know that raising A to a positive power will equal the right side with D raised to that same power. i.e. A^n = P^-1 * D^n * P

Does this imply that if T is diagonalizable so is T^2? what if T^2 is diagonalizable, then does that imply T is diagonalizable?

I think that it is an iff statement. Can anyone shine some light into this situation? maybe a counter example? is there a way to generalize the relationship between T and T^2?

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# Way to generalize the relationship between T and T^2?

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