I was studying the Stephen Boyd's textbook on convex optimization and have a question. The book says the following:
The singular values of $A$, $\sigma_1 \geq \sigma_2 \geq ... \geq \sigma_n$ are the square roots of the eigenvalues of $\lambda_1 \geq \lambda_2 \geq ... \geq \lambda_n$ of $G$ ($G$ is the Gram Matrix, $A^T A$). Therefore $\sigma_1^2$ is a convex function of $G$, and $\sigma_n^2$ is a concave function of $G$.
Can anybody explain why the maximum and minimum eigenvalues are a convex and concave function of a Gram matrix?