Is A^k a Projection Operator if k is Even/Odd?

[LagrangeEuler]

If $A$ is not projection operator. Could $A^k$ be a projection operator where $k$ is even or odd degree. Thanks for the answer.

 

[hilbert2]

A trivial example is the matrix $A=\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$, the square of which is the identity matrix. The identity operation is technically a projection.

 

[LagrangeEuler]

I need projection matrix $P$ on the direction. So matrix for which eigenvalues are $1$ and $0$, and for which $TrP=1$.