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?



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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.