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Quick clarification on the following will be appreciated.

I know that for a real symmetric matrix $M$, the maximum of $x^TMx$ over all unit vectors $x$ gives the largest eigenvalue of $M$. Why is the "symmetry" condition necessary? What if my matrix is not symmetric? Isn't the maximum of $x^TMx=$ still largest eigenvalue of $M$?

Thanks.

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    As counterexample you can consider $M = \left( \begin{array}{c c} 0 & 1 \\ 0 & 0 \end{array}\right)$2011-12-01

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