TITLE: Find the eigenvalues of a projection operator
QUESTION [26 upvotes]: A projection operator $P$ is defined as $P^2$=$P$. Use this definition to find the eigenvalues of this operator.

In this question is it necessary to define what the projection operator is? And won't the eigenvalue just be zero?

REPLY [10 votes]: The eigenvalues are $0$  and $1$. Indeed, we know that for a projector $P$ defined on a vector space $E$, we have
$$ E= \ker P\oplus \operatorname{im}P=\ker P\oplus \ker(I-P) $$
$\ker P$  is the eigenspace associated with the eigenvalue $0$, $\ker(I- P) $  the eigenspace associated with the eigenvalue $1$. As $E$ is the direct sum of these eigenspaces, we have all eigenspaces, and all eigenvalues.