Projection operator

If ##A## is not projection operator. Could ##A^k## be a projection operator where ##k## is even or odd degree. Thanks for the answer.
 
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hilbert2

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

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