Linear Algebra - Normal and Unitary Matrices

[estro]

Homework Statement

Suppose V is a unitary space [over C] and T: V -> V is a normal transformation that satisfies T^-1=-T. Prove that T is unitary transformation.

Homework Equations

I know that T is unitary if and only if it is normal and the absolute value of its eigenvalues is 1. [*2]

The Attempt at a Solution

T^-1=-T so T²=-I, now suppose a is an eigenvalue of T so T²v=a²v=-Iv what in turn means $a=\pm 1$.
So from [*2] we can conclude that T is unitary.

Have I missed something?

 

[Kreizhn]

Did you prove normality? Also, can you be more specific about V? Is it just a vector space? A Banach Space? Hilbert Space? Is it finite or infinite dimension? What definition of unitary are you using?

 

[estro]

Thanks for posting!

Normality is given.
V is a general unitary space. [it means V is a general space over the complex field]
Forgot to mention: V is finite dimension space.
Unitary definition:$T \cdot T^*=T^* \cdot T=I$

 

[Kreizhn]

Ah yes, I see now that it was given. It looks fine to me, though I have to say I've never heard of a "general unitary space." Perhaps this is okay in this instance since $\mathbb C$ is algebraically self-dual, but normally one requires at the very least a normed vector space.

 

[estro]

So I guess the prove is ok? [This question was formulated by myself so it is totally possible that I didn't define the question itself well]

What is normed vector space?

 

[Kreizhn]

It looks fine so long as the result about being normal with modulus 1 is true. I think it's true, but haven't run through the proof myself. In any case, I would say you're good to go.

A normed vector space is a vector space with a norm. :) Not sure I can be more explicit than that.

 

[estro]

It looks fine so long as the result about being normal with modulus 1 is true. I think it's true, but haven't run through the proof myself. In any case, I would say you're good to go.

A normed vector space is a vector space with a norm. :) Not sure I can be more explicit than that.

Thanks!

...
A normed vector space is a vector space with a norm. :) Not sure I can be more explicit than that.

I though every vector space has a norm, so I should not explicitly say it.