Square Matrix A that is not Diagonalizable but A^2 is Diagonalizable

[Deneb Cyg]

The question is "give an example of a square matrix A such that A^2 is diagonalizable but A is not."

I know that if A^2 is diagonalizable, A^2 = P(D^2)P^-1. And if A is not diagonalizable, there is no invertible matrix P and diagonal matrix D such that A=PDP^-1.

However I'm not sure how to begin finding an example.

Can someone explain how to go about finding this?

 

[lanedance]

I'm assuming you're talking about a matrix over the reals...

The diagonalising matrix P is composed of eignevectors, so the eigenvectors need to exist and span dimension n, for the nxn matrix diagonlised form to exist

so you could try and look for a matrix A where no real eigenvalues exist... but A^2 has real eigenvalues...

2x2 is a good place to start...

 

[HallsofIvy]

And i and -i are good eigenvalues to start with!