Suppose we have a real matrix $A$ which satisfies $A^4=I$, can we determine if $A$ is diagonalizable?
I believe the answer is that we can't because all we know about the matrix $A$ is that it is invertible (otherwise $A^4$ couldn't be an invertible matrix)..
How can I prove it? How can I find such a matrix $A$ which isn't diagonalizable but $A^4 = I$?
The only matrix $A$ I was able to find which satisfies $A^4=I$ is the identity matrix itself but the identity matrix is diagonalizable.
Best Answer
No, consider the rotation by $\pi/2$ in $\mathbb R^2$: $$A=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$$ Which must satisfy $A^4=I$, but has no (real!) eigenvectors, so it's not diagonalizable.