Let Vbe an m-dimensional subspace of R^n. I have already found the eigenvalues of the nxn orthogonal projection matrix A onto V as 0 and 1 with respective eigenspaces V_perp. and V, and dimensions of m and n-m. Also, I found the eigenvalues of the nxn reflection matrix B "across" V as 1 and -1 with respective eigenspaces also of V and V_perp., also with dimensions m and n-m.
Out of that I'm certain of most except the n-m dimensions.
But now I have "What can you say about the eigenvalues' algebraic and geometric multiplicities?" I know that gemu(A, B) <= almu(A,B) and that A and B are diagonalizable iff sum(gemus(A,B))=n.
Unfortunately I dont see how these help me. I'm trying to think about the problem with n=3, and m=2, eg, for reference but I don't want to extrapolate.
Pls no extremely complicated answers: young'un here.
Advance thanks