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Q:
Correspondence between projections and line spectrum
Let $X$ be a finite dimensional inner product space. Then a projection $P$ is a bounded linear operator on $X$ so that $P^2 = P$. In the case of an inner product space, this means that $P$ is self-adjoint, which means that $P = P^*$.
What then is the corrrespondence between projections and the spectrum of $P$? Namely, what would the possible values of $\lambda$ be that correspond to eigenvectors of $P$?
A:
For each $\lambda\in\mathbb C$ the equation $$Px=\lambda x$$ has a nonzero solution $x$, so $\mathbb C\setminus\{0\}\subseteq\sigma(P)$, and $0\in\sigma(P)$ iff $P$ is not invertible.
You can verify all the other properties by noting that $P-\lambda1$ is a nonzero projection on $X$ and hence invertible.
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