Stuck on step in a linear algebra proof

In summary: First, T*v is just a vector, so you can pull it out of the inner product:<Tv,Tv> = T*<v,Tv> In summary, the conversation discusses the definition of a normal operator and the proof for the equivalence stating that an operator T is normal if and only if ||Tv|| = ||T*v||. The conversation also mentions the use of inner products and properties of operators to prove the theorem.
  • #1
ocelotl
1
0

Homework Statement


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.

Homework Equations


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.
 
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  • #2
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:

1. What is the definition of a linear algebra proof?

A linear algebra proof is a mathematical argument that uses various properties and theorems of linear algebra to demonstrate the validity of a statement or equation.

2. How do I know if I am stuck on a step in a linear algebra proof?

If you have been working on a proof and cannot seem to make progress, or if you are unsure about the validity of a particular step, then you may be stuck on a step in your linear algebra proof.

3. What should I do if I am stuck on a step in a linear algebra proof?

If you are stuck on a step in a linear algebra proof, it is helpful to review the definitions, properties, and theorems that you have used so far. You can also try approaching the problem from a different angle or seeking help from a classmate or professor.

4. How can I avoid getting stuck on a step in a linear algebra proof?

To avoid getting stuck on a step in a linear algebra proof, it is important to have a strong understanding of the fundamentals of linear algebra and to carefully plan your approach before starting the proof. It is also helpful to regularly practice solving proofs.

5. Are there any common mistakes that people make when stuck on a step in a linear algebra proof?

Yes, some common mistakes that people make when stuck on a step in a linear algebra proof include using incorrect definitions or theorems, making algebraic errors, and misunderstanding the problem or the steps needed to solve it. It is important to carefully check your work and approach the problem systematically to avoid these mistakes.

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