$P$ is a stochastic matrix i.e. square, non-negative, rows sum to one. Let $\Phi$ be a real matrix of size $n \times k$ with independent columns and $k < n$. Let $\Xi$ be the diagonal matrix with a principal left eigenvector of $P$ on the diagonal (for ergodic $P$s, the stationary distribution). Let $\| \cdot \|_2$ be the matrix norm $L_2$.
I am trying to prove the following.
$\| \Phi^\top \Xi P \Phi \|_2 \leq \| \Phi^\top \Xi \Phi \|_2 \quad \quad$
My attempts at the solution this far are given below.
I can solve the case when $\Phi = I$ as follows.
$\| \Xi P \|_2 \leq \sqrt{\| \Xi P \|_1\| \Xi P \|_\infty} = \sqrt{\| \Xi \|_2 \|\Xi\|_2} = \|\Xi\|_2$ The first equality holds because $\| \Xi P \|_1 = \| \Xi P \|_\infty = \| \Xi \|_2$, which can be shown by plugging the definitions of the norms.
Unfortunately, I realized that the property $\forall \Psi, A, B. \left[ \|A\|_2 \geq \|B\|_2 \Rightarrow \|\Psi^\top A \Psi\|_2 \geq \| \Psi^\top B \Psi \|_2 \right] $ doesn't hold, which I had hoped might help me solve the problem.
Maybe there is some weaker property that I could use, which does hold?
If you can't contribute a full proof, I would be grateful if you could suggest proof techniques or hints I could try, since I am new to this sort of thing.