For symmetric matrices $M$ and $N$, it is easy to verify that $\sigma_{max}(M)\leq\sigma_{min}(N)$ is a sufficient condition for $M\leq N$, where $\sigma_{max}(.)$ and $\sigma_{min}(.)$ denote the maximum and minimum singular values.
I wanted to prove if it is also a necessary condition.
In other words, is it true that $M\leq N$ if and only if $\sigma_{max}(M)\leq\sigma_{min}(N)$?