Let two vectors, $\mathbf{x}, \mathbf{y}$ be related as: $0 \leq x_i \leq \lambda y_i$, for some $\lambda > 0$. That is, $\mathbf{x}$ is coodinate-wise dominated by a scaled version of $\mathbf{y}$.

Also let $\mathbf{M}$ be a positive semi-definite matrix. My question is, how are their matrix $\mathbf{M}$ induced weighted norms related? I think they are related as: $x\mathbf{M}x' \leq \lambda^2 y\mathbf{M}y'$. Yet I am so far unable to prove this. How do I prove (if above relation is not correct, then a correct version) this relation?