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

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  • $\begingroup$ Try the square root of the maximum eigenvalue of $G^*G$. $\endgroup$ – copper.hat Dec 9 '13 at 16:11
  • $\begingroup$ I am a little bit confused... Maximum eigenvalue of G or $G^TG$? $\endgroup$ – Niousha Dec 9 '13 at 16:34
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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 \begin{equation} \max_{\lambda \,\,\in \,\,eigval(G)} |\lambda| \end{equation} namely, the numerical value here is the maximum of the absolute value of any eigenvalue of $G$.

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  • $\begingroup$ I see. So I think the minimum value of y is zero, in case u is an eigenvector of $G^TG$. Right? $\endgroup$ – Niousha Dec 9 '13 at 16:20
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Hint: Try Cauchy-Schwarz inequality

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