Unitary matrix eigenvalues

[mooeypoo]

Hey again, I'm dealing with QM and the subject is entirely new to me. I'm having troubles figuring out how to start the questions. Here's another in my frustrated series:

 

Problem 2:

If U is a unitary matrix, show that the absolute value of every matrix element is less than or equal to 1, ie:

[math]|U_{ij}|\leq 1[/math]

 

So in this case, here's how I start my solution (couldn't find the 'cross' symbol, so I'll use * instead):

 

U is a unitary matrix, so [math]U^{*}U=UU^{*}=I_n[/math] and [math]U^{*}=U^{-1}[/math]

 

And so, I started:

 

[math]UX = \lambda X[/math]

[math]U^{*}UX=U^{*}\lambda X=\lambda U^{*} X[/math]

[math]I_{N}X=\lambda U^{*} X[/math]

 

And I am thinking that since I_{N} is the identity matrix, if I can find a way to show that lambda is smaller or equal to that, then I finish the proof. But... I ... don't... see how...

 

I'm stuck! help!

 

~moo

[ajb]

What you have shown is that if [math]U X = \lambda X[/math] then [math]U^{*}X = \frac{1}{\lambda} X[/math] i.e. it is also an eigenvector of the [math]U^{*}[/math] with the eigenvalue being the inverse of [math]\lambda[/math].

 

Now take inner products of the above expressions with [math]X[/math], from the left with the first expression and from the right with the second.

[timo]

EDIT: Ah well, just do some work and follow ajb's proposal (assuming it works out).

Just in case that is not clear: showing that [math] |\lambda |=1 [/math] is not what the question asked for.

Edited September 25, 2010 by timo

[ajb]

Maybe I have miss read it. sorry

 

I wonder if taking the determinant will help?

 

We know [math]|det U|^{2} = 1[/math].

Edited September 25, 2010 by ajb

[timo]

I was hoping that Moo had an idea how to do the proof once she had proven that [math]|\lambda |=1[/math]. If that was not the case then here's a hint how I'd solve it: [math]U^* U = 1[/math] by definition and [math]U^* U[/math] contains the sum of several products of the [math]U_{ij}[/math] (and their conjugates). But you (=Moo) should first try your idea, in case you have one.

[ajb]

how I'd solve it: [math]U^* U = 1[/math] by definition and [math]U^* U[/math] contains the sum of several products of the [math]U_{ij}[/math] (and their conjugates).

 

That sounds very inelegant, but works. My other idea was to write out the determinant, this would also be very inelegant and far harder than what you suggest.

 

Is there some clever argument to use here?

[timo]

My suggestions works reasonably well if you write it down in index notation. My first idea was to select a single [math]U_{ij}[/math] by sandwiching the matrix with two base vectors (of the basis the matrix is written in) and then compare it to the result you get when the base vectors are are represented as a combination of eigenvectors (thereby indeed using [math]|\lambda |=1[/math]. You might call that elegant but I'm not sure it is faster - I didn't even try if it works.

[ajb]

My suggestions works reasonably well if you write it down in index notation.

 

Yes of course.

 

My first idea was to select a single [math]U_{ij}[/math] by sandwiching the matrix with two base vectors (of the basis the matrix is written in) and then compare it to the result you get when the base vectors are are represented as a combination of eigenvectors (thereby indeed using [math]|\lambda |=1[/math]. You might call that elegant but I'm not sure it is faster - I didn't even try if it works.

 

I think that may work. Honestly, I can't think of anything other than brute force in index notation.

[mooeypoo]

I'm very confused, I have no idea what you are talking about... :\ am I even at the right track, or should I try what you propose... I.. meh. I'm lost.

[timo]

I don't see how your approach is going to help you. You need a statement about the entries of U, the [math]U_{ij}[/math]. So at some point you must either separate out a single matrix entry (sketched in post #7) or obtain a set of equations of these entries (post #4 and #5 are two such approaches) that allow to make the desired statement. Unless it is important that you do this homework assignment I don't think I should give any more hints; there's probably not much learning involved in writing down a proof when you were told all of the steps. And don't worry if you didn't get the communication between Andrew and me - you were not supposed to.

[mooeypoo]

I already gave the homework assignment with my partial work, so it's not about the grade anymore. I'm simply confused.

 

I'm going to sit down and just try this out from scratch again, I think my confusion is confusing me even more, if that makes any sense. The way things are written out in this course is very confusing, specially the dirac notation and the bra-ket, so I think I'm overthinking everything and confusing myself.

 

I'll just have to sit down and redo this. Thanks for the help so far I'll post when I finish, which will probably be in the weekend when I'm not as tired over other classwork (that probably contributes to my confusion, too..)

 

Anyways, thanks for the help so far.

[timo]

Don't waste time on it if you've got better things to do. It's just a math proof for a statement I think I've never needed; no physics to learn there.