Are Eigenvectors of Unitary Transformations Orthogonal?

[Ed Quanta]

Homework Statement

Show that the eigenvectors of a unitary transformation belonging to distinct eigenvalues are orthogonal.

Homework Equations

I know that U+=U^-1 (U dagger = U inverse)

The Attempt at a Solution

I tried using a similar method to the proof which shows that the eigenvectors of hermitian transformations belonging to distinct eigenvalues are orthogonal.

So assume our eigenvectors are a and b. I assumed U(a)=xa and U(b)=yb

x<a|b>=<Ua|b>=<a|U^-1b>= ?

Help anyone. I know this probably isn't too rough.

 

[Dick]

If U(a)=xa and you act on both sides with U^(-1), what does that say about eigenvectors of U^(-1)?

 

[Ed Quanta]

They are the reciprocals. U^(-1)a=1/x

 

[Ed Quanta]

So

x<a|b>=<Ua|b>=<a|U^-1b>=<a|(1/y)b>=1/y<a|b>

So (x - 1/y)<a|b>=0

Now how do I know x - 1/y cannot equal 0?

 

[Dick]

Be a little careful. You are probably dealing with a complex inner product. If it's real then this is fine. As U is orthogonal, what do you know about the absolute value of x and y?

 

[Ed Quanta]

The absolute values of x and y must be real.

 

[dextercioby]

HINT: The spectrum of a unitary operator in a complex Hilbert space is the unit circle...

 

[Dick]

<a,b>=<Ua,Ub>. Apply that to an eigenvector. As dextercioby says...

 

[Ed Quanta]

Ok, so I get that the norm of the eigenvalues must equal 1.

<a|b>=<Ua|Ub>=x*y<a|b>

x*y=1?

 

[Dick]

As I've said, be a little careful. You are correct in the case if U is real. But if U is complex, the condition is x^* y=1. So if x=y, then x^* x=1 and the eigenvalues are unit complex numbers. How does this help you with the original problem?

 

[Ed Quanta]

So x*y does not equal 1 unless y=x.

 

[Dick]

So x*y does not equal 1 unless y=x.

If you mean x and y being real numbers with norm 1, then yes.

 

[Ed Quanta]

I'm still confused man. I want to show that x*y<a,b>-<a,b>=0

I want to show then that x*y does not equal 1. Where do the norms fit in?

 

[Dick]

So

x<a|b>=<Ua|b>=<a|U^-1b>=<a|(1/y)b>=1/y<a|b>

So (x - 1/y)<a|b>=0

Now how do I know x - 1/y cannot equal 0?

You've gotten this far and have assumed x and y are DIFFERENT eigenvalues of U. If U is real this is super easy, since x and y are both in the set {+1,-1} and DIFFERENT. What about U complex? Then you have to mend your ways and remember &lt;c x,y&gt;=c^*&lt;x,y&gt;.