Find eigenvalues, kernel and Image of an Orthogonal projection

Question

Let $V$ be a Vector space with inner product and $U$ a subspace. Let $P$ be the orthogonal projection over $U$. Find eigenvalues, kernel and Image of $P$.

I know I have to consider the special cases of $U=\{0\}$ and $U=V$. But I don't understand How to make the general case.

Answer 1

"An orthogonal projection has spectrum $\,\sigma(P)=\{0,1\}$, if $P$ is neither zero ($\iff\ker P=V)\,$ nor equal to the identity ($\iff\ker P=\{0\}),\:\ldots\,$"
could become a complete answer, but.

Searching this site for $\:$ projection kernel image $\:$ yields many$^\text{many}$ hits, amongst them the following sorted-by-age-almost-dozen, and you are invited to browse thru:

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