Let P = the map which takes to its orthogonal projection onto S, a subspace of an inner product space V.

dimS = k, dimV = n

Find the eigenvalues and eigenvectors of P. What are the algebraic and geometric multiplicities of each eigenvalue?

Ok so what I know is that P:S-->S(orthonormal projection), and that dimS(orthonormal projection) = n-k. But how would I make this into a matrix where I can find the eigenvalues and eigenvectors?

My guess would be to say P is a projection of a vector x in S onto S(orthonormal projection) which means P(x) = SUM(<x, si> si) where si is a vector in S(orthonormal projection). Then I would say that [x]_s = (<x,s1>, <x,s2>,...,<x,sn>), but then I don't know how to find the eigenvalues of a matrix that isn't square.