Norm in Matrix algebra

**Mauritzvdworm** · July 14th 2010, 12:37 AM

let $A\in M_n(\mathbb{C})$ , show that
$\|A\|^2=\max\{\lambda: \det(\lambda-A^*A)=0\}=\max\{\lambda: \det(\lambda-AA^*)=0\}$

**Opalg** · July 14th 2010, 08:39 AM

Originally Posted by Mauritzvdworm

let $A\in M_n(\mathbb{C})$ , show that
$\|A\|^2=\max\{\lambda: \det(\lambda-A^*A)=0\}=\max\{\lambda: \det(\lambda-AA^*)=0\}$

The proof consists of two steps. (1) $\|A\|^2 = \|A^*A\|$ ; (2) A*A is a positive definite matrix and so its norm is equal to its largest eigenvalue. Those two steps together show that $\|A\|^2=\max\{\lambda: \det(\lambda-A^*A)=0\}$ .

For the last part, use the fact that $\|A\| = \|A^*\|$ to deduce that $\|A\|^2=\max\{\lambda: \det(\lambda-AA^*)=0\}$ .

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