TITLE: Maximal eigenvalue is a convex function. Why?
QUESTION [6 upvotes]: Let $A$ be a symmetric real matrix. Let $f(A)=\lambda_{\max}(A)$ be its largest eigenvalue. Why is $f$ convex?

REPLY [11 votes]: $f(A) = \sup_{\|x\|=1} x^T Ax$, and a supremum of convex functions is convex.
(For a given $x$, the function $A \mapsto x^TAx$ is linear, hence convex.)