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
and since eigenvectors are the vectors which satisfy with the matrix of your linear transformation and an eigenvalue of , 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.