Find the eigenvalues and eigenvectors of A geometrically.
A=
[1 0
0 1]
(projection onto the x-axis)
the matrix for the projection on the x axis is
$\displaystyle
\left( \begin{array}{ccc} 1 & 0 \\
0 & 0
\end{array} \right)
$
and since eigenvectors are the vectors $\displaystyle x$ which satisfy $\displaystyle Ax={\lambda}x$ with $\displaystyle A$ the matrix of your linear transformation and $\displaystyle {\lambda}$ an eigenvalue of $\displaystyle A$, the eigenvectors of the projection on the x axis must be all the vectors that are already on the x axis and have an eigenvalue of 1. you can also see this by observing the matrix which is already diagonnal and gives you right away its only eigenvalue 1.