maximum and minimum singular values

Question

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?

Thanks.

Answer 1

If $A$ is a real symmetric matrix, then its maximal eigenvalue is $$\lambda_1(A)=\max_{\|v\|=1} \langle Av, v\rangle \tag1 $$ and its smallest eigenvalues is $$\lambda_n(A)=\min_{\|v\|=1} \langle Av, v\rangle \tag2 $$ For each fixed vector $v$ the function $A\mapsto \langle Av, v\rangle$ is a linear function of $A$. The maximum of any family of linear functions is convex. The minimum of any family of linear functions is concave.