# Counterexample for $\| (\Phi^\top \Xi \Phi)^\dagger \Phi^\top \Xi P \Phi \|_2 \leq 1$

All matrices are real.

Let $P$ be any stochastic matrix (i.e. square, non-negative, rows summing to one).

Let $\Xi$ be a diagonal matrix containing on its diagonal any left eigenvector of $P$ corresponding to eigenvalue one.

Let $\Phi$ be any matrix (we can assume independent columns, but I don't think this is necessary). Typically, $\Phi$ has much less columns than rows.

Let $\| \cdot \|_2$ denote the matrix norm induced by the vector norm $L_2$.

Prove the following statement:

$\| (\Phi^\top \Xi \Phi)^\dagger \Phi^\top \Xi P \Phi \|_2 \leq 1$

In the above $(\cdot)^\dagger$ denotes the Moore-Penrose pseudoinverse. Note that for ergodic $P$s, it becomes equivalent to the ordinary inverse.

In case this is not true, please provide a counterexample.

Even if you could help me just with the ergodic case, I'd be really grateful.

Edit: thanks to @loupblanc, I discovered that my statement was actually incorrect.

• Are they no conditions on phi like norm phi <= 1? – TenaliRaman Nov 8 '14 at 15:53
• @TenaliRaman the norm of $\Phi$ doesn't matter because $\Phi$ occurs twice in the inverse and twice outside (i.e. you can cancel). – ziutek Nov 8 '14 at 16:00
• Is the diagonal of $Ξ$ a LEFT eigenvector of $P$ ? – loup blanc Nov 8 '14 at 19:43
• @loupblanc Yes it is a left eigenvector (i.e. the stationary distribution if $P$ is ergodic). Sorry for being unclear -- I have now edited the question. – ziutek Nov 9 '14 at 11:41

Take $\phi=\begin{pmatrix}86&38\\-51&-38\\51&-19\end{pmatrix},P=\begin{pmatrix}0.32&0.55&0.13\\0.58&0.05&0.37\\0.73&0.09&0.18\end{pmatrix}$. I find $||.||_2\approx 2.35$.