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The norm is defined as $\|A\|=\sup\{ \|A v \| : \|v\|=1\}$. I want to show it is equal to the square root of the largest eigenvalue of $A^tA$.

I do not know why it is an eigenvalue of a product of $A^tA$ not simply an eigenvalue of A. How to proceed?

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    You write $^*$, then transpose: are you working with real of complex matrices?2012-10-14
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    Are you sure your definition is written as you wanted? First, why use $A^*A$ and not simply $A$? Because $A$ might have complex eigenvalues! Second, you want to write $\| A \|=sup\{\| A^*A v \colon \| v \|=1 \}$ and in this case this is a well-known variational characterization of largest eigenvalue.2012-10-14

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