1. ## Eigenvalue, norm

Hello!

I've got a question according the solution of this exercise

$A \in M(n, \mathbb{K})$

Show that $\|A \| := \sqrt{E_{max}}$, where

$E_{max}$ is the biggest eigenvalue $of A^*A$

and

$\| A \| := \sup_{\| x \|_2 = 1} \| Ax\|^2 = \sup_{\| x\| \le 1} \langle Ax, Ax \rangle$

$= \sup \langle A^* Ax,x \rangle$

SOLUTION

$A^*A$ is positive definit, so there is a unitary $U \in GL(n, \mathbb{K}) :$

$UA^*AU^{-1} = diag (E_1, ... , E_n )=" alt="UA^*AU^{-1} = diag (E_1, ... , E_n )=" /> and $E_i \ge 0$ denote the eigenvalues of $A^*A$

So

$\| A \|^2 = \sup \langle A^*Ax, x \rangle$

$= \sup \langle U^{-1}DUx,x \rangle$

$= \sup \langle DUx , Ux \rangle , \ Ux =: y$

$=\sup \langle Dy, y \rangle$

$= \sup \sum^n_{i=1} E_i |y_i |^2$

$= E_{max}$

So why is the last line? Why is it not $\ge E_{max}$?

Rapha

2. That is because $y = Ux$ is supposed to be a unit vector, so $\sum^n_{i=1} |y_i |^2 = 1$. Therefore $\sum^n_{i=1} E_i |y_i |^2\leqslant \sum^n_{i=1} E_{\max} |y_i |^2\leqslant E_{\max}$.

3. Hello Opalg!

Thanks alot.