# Thread: Norm in Matrix algebra

1. ## Norm in Matrix algebra

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

2. 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 $\|A\|^2=\max\{\lambda: \det(\lambda-AA^*)=0\}$.