# $L_2$ norm and matrix norm inequalities

Given a semi-positive definite matrix $A \in \mathbb{R}^{n \times n}$ and $x,y \in \mathbb{R}^n$ such that: $$\left\| Ay\right\|_2 \leq \left\| Ax\right\|_2 \leq \sqrt{n} \cdot \left\| Ay\right\|_2$$

Is it true to say that: $$\left\| x\right\|_2 \leq \sqrt{n} \cdot \left\| y\right\|_2 \ ?$$

Let $x$ and $y$ be $\lambda_1$- and $\lambda_2$-eigenvectors of $A$ respectively, and let $y$ have unit norm. Then the condition becomes $$\lambda_2 \le \lambda_1 \|x\|_2 \le \sqrt{n} \lambda_2$$ and you want to deduce $$\|x\|_2 \le \sqrt{n}.$$
From here it is easy to provide a counterexample: let $\lambda_1=n$, $\lambda_2=n^2$, and $\|x\|_2=n$.