# eigen values and eigen vectors of the projection

Assume that $W$ is n-dimensional subspace of an m-dimensional vector space $V$. Find all eigenvalues and all eigenvectors of the projection operators $P_W$.

Here is my ideas:

Since $W$ is n-dimensional subspace of an m-dimensional vector space $V$, then $dim W <dim V$. Suppose $\lambda \in F$ is an eigenvalue of $P_W$. Then there exists non zero vector $w$ such as $P_W w = \lambda v$. Suppose $w \in W$ are linear independent, then $$w = \lambda _1 w_1 + ...+ \lambda _m w_m = 0$$

Then we have the eigenvalues of zeros, and How do i find eigen vectors?

Thanks!

When talking about the projection, not only you need the space on which you project, you also need the space that is used as a direction of projection. Imagine the projection in $\Bbb R^2$ on the subspace $\{y=0\}$ along the subspace $\{x=0\}$. It it not the same as the projection on the $\{y=0\}$ along the subspace $\{x-y=0\}$.
In you case - decompose the space $V=W\oplus X$ such that $\ker P_W = X$ and $im\, P_W=W$, $P_W^2=P_W$. The latter equation is a necessary and a sufficient condition that $P_W$ is a projector. This equation also implies that possible eigenvalues of $P_W$ are $0$ and $1$.
Consider any vector $w\in W$, then $P_Ww=w$, therefore any nonzero vector in $W$ is an eigenvector. Similarly, if $v\in X$, then $P_Wx=0$, and $x$ is also an eigenvector.