# Homework Help: Stuck on step in a linear algebra proof

1. Sep 4, 2011

### ocelotl

1. The problem statement, all variables and given/known data
Axler's Linear Algebra done right, propositions 7.6 states:
An operator T is normal if and only if ||Tv|| = || T*v||, T* being the adjoint of T.

I've been stuck on the last equivalence in the proof stated here:

<T*Tv,v>=<TT*v> <=> ||Tv||^2=||T*v||^2, with squaring both sides of the second statement proves the theorem.

2. Relevant equations
3. The attempt at a solution
Basically all my attempts have been stopped by the fact that i can't either get rid of T or T* in the second slot when working backwards or trying to get T or T* back when working through it forwards. Any help would be highly appreciated.

2. Sep 4, 2011

### lanedance

so i take it you definition of normal is:
T*T = TT*

and the problem you are having is:
Assume ||Tv|| = || T*v||, show T is normal

Now assuming all that is the case you squaring the relation is a simple matter
||T*v||^2 = ||Tv||^2

Then writing in terms of inner products you have
<T*v,T*v> = <Tv,Tv>

now i would use the properties of operators in inner products to ove an operator within the inner product, so both act on a single vector within the inner product

Last edited: Sep 4, 2011