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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.

NuKexZ
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  • Try to visualize this in two dimensions where $U$ is the x-axis. Can you see the kernel (i.e. which vectors are projected to $0$) and the range? Then generalize.

    As for the eigenvalues it depends on what you've learned. This is easy if you've learned about the minimal polynomial of an operator. Otherwise you could find a good basis of $V$ where $P$ has a simple form.

     – Olivier Moschetta Nov 26 '18 at 14:13
  • What is the image of an element of $U$? What is the image of an element of $U^\perp$? The answers give you two possible eigenvalues for $P$. – amd Nov 26 '18 at 23:32

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"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:

Hanno
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