Question

It may be better to post this question here. Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric positive-semi-definite matrix with exactly $m/2$ positive eigenvalues and every entry of $M$ is between $0$ and $1$. Let $I_{r}$ be an $m$ by $m$ diagonal matrix with $m/2$ of $1$'s on its diagonal and other diagonal entries being zeros (different arrangements of $1$'s and zeros on the diagonal give different $r$). Would $m^{2}/\inf_M\sup_{r}\left(\left\Vert \left(I_{r}MI_{r}\right)^{\dagger}\right\Vert \right)$ be bounded? where $X^{\dagger}$ is the extended pseudoinverse of $X$, defined as $\left(U\mbox{diag}\left(d_{1},\ldots,d_{m/2},0,\ldots,0\right)U^{T}\right)^{\dagger}=U\mbox{diag}\left(d_{1}^{-1},\ldots,d_{m/2}^{-1},0,\ldots,0\right)U^{T}$ in which $d_{i}^{-1}=\infty$ if $d_{i}=0$. Thanks.