# Optimization problem: Finding the maximum value

Can someone please give me a hint on this problem? I want to find the maximum value of y, given the equations:

• Try the square root of the maximum eigenvalue of $G^*G$. – copper.hat Dec 9 '13 at 16:11
• I am a little bit confused... Maximum eigenvalue of G or $G^TG$? – Niousha Dec 9 '13 at 16:34

Assuming that $\|\cdot\|$ denotes the $2$-norm, the quantity you are trying to maximize is the induced norm of the matrix $G$ (assuming as well that $G$ is meant to be a matrix and $u$ a vector of compatible dimension). Since you are maximizing the square of the induced norm of $G$, this will give you the square of the largest singular value of $G$. If $G$ has full rank, it will give you $$\max_{\lambda \,\,\in \,\,eigval(G)} |\lambda|$$ namely, the numerical value here is the maximum of the absolute value of any eigenvalue of $G$.
• I see. So I think the minimum value of y is zero, in case u is an eigenvector of $G^TG$. Right? – Niousha Dec 9 '13 at 16:20