Q:

Norm of matrix and its maximum eigenvalue

I've seen in some inequalities in the theory of ODEs that $\lVert Q \lVert \le \lambda_{max}(Q)$. What theorem from Linear Algebra is relevant here?

A:

I wonder whether the inequality isn't backward. 
$\|A\|_2$ is usually defined as the supremum, over all unit vectors $x$, of $\|Ax\|_2$. 
Let $\lambda$ be the largest eigenvalue of $A$ (in modulus), and let $v$ be a corresponding unit eigenvector. Then $$\|A\|_2\ge\|Av\|_2=\|\lambda v\|_2=|\lambda|\|v\|_2=|\lambda|$$

