$$\rho =v{v}^{\ast},$$

where v is a $n\times 1$ unit vector.

Let M be a positive semidefinite matrix such that all its eigenvalues are between 0 and 1.

I am trying to see whether the following two inequalities are correct:

$$\text{Tr}\left({M}^{2}\rho \right)\le \text{Tr}\left(M\rho \right).$$

$$||Mv||\le \text{Tr}\left(M\rho \right),$$

where $||\cdot ||$ is the 2-norm of a vector.