I am looking for a proof using the min-max principle. Wikipedia seem to provide just that: http://en.wikipedia.org/wiki/Min-max_theorem#Cauchy_interlacing_theorem
But this part seems to be wrong:
This can be proven using the min-max principle. Let $\beta_i$ have corresponding eigenvector $b_i$ and $S_j$ be the $j$ dimensional subspace $S_j=\operatorname{span}\{b_1,\dots, b_j\}$, then $ \beta_j = \max_{x\in S_j,\|x\|=1}(Bx,x) =\max_{x\in S_j,\|x\|=1}(PAPx,x) =\max_{x\in S_j,\|x\|=1}(Ax,x)$
How is the shift from $PAPx$ to $Ax$ legal? $PAP$ is an $m\times m$ matrix while $A$ is an $n\times n$ matrix. $x$ can't fit both. Can anyone correct the proof?