TITLE: Eigenvalues bounded $\implies$ Matrix entries bounded?
QUESTION [6 upvotes]: Suppose $H$ is a Hermitian matrix with all its eigenvalues be bounded (below and above) say between $-n_0$ and $n_0$. Is it true that all members of the matrix $H$ is bounded (below and above) by some constant which does not depend on the order of the matrix?

REPLY [10 votes]: Since $-n_0 I \le H \le n_0 I$ (in the sense that $H + n_0 I$ and 
$n_0 I - H$ are positive semidefinite), $\|H\| \le n_0$ and
the matrix entries $|a_{ij}| = |e_i^T H e_j| \le \| H\| \le n_0$.