If lambda max is the largest eigenvalue of a real symmetric matrix A, show that no diagonal entry of A can be larger than lambda max.
First, show the property about $\displaystyle \lambda_{\max}$ I mentioned when $\displaystyle A$ is a diagonal matrix. Jump to the general case using the fact that there exists an orthogonal matrix $\displaystyle P$ and a diagonal matrix $\displaystyle D$ such that $\displaystyle P^tDP=A$.