Unitary Operators: Why is Spectrum on Unit Circle?

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In summary, the spectrum of a unitary operator in a Hilbert space is the set of eigenvalues, all of which lie on the unit circle. This is due to the unitariness of the operator and the fact that its eigenvectors have unit length. The length of Ux is equal to the inner product of an eigenvector with its eigenvalue, and this can be pulled out as a number multiplier based on a formula. This concept is important to understand when working with unitary operators in a Hilbert space.
  • #1
Raven2816
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Homework Statement



why is the spectrum of the unitary operator the unit circle?

Homework Equations



i know that U^(-1)=U* and i know this makes U normal
i also know that normal means UU*=U*U


The Attempt at a Solution



i know that from spectral theory there is some lambda in the spectrum
such that abs(lambda)=1, but i don't understand why ALL of them are on
the unit circle. (i understand the operator, but spectrums are confusing to me.

thanks
 
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  • #2
If x is an eigenvector of U, and [itex]\lambda[/itex] is its eigenvalue, then what is the length of Ux?

[itex]Ux = \lambda x[/itex]
 
  • #3
hmmm, i know that, so i have Ux = Lx and L is 1...so then Ux = x...? I'm just getting lost
 
  • #4
Just so we're happy: you're only talkiong about operators on finite dimensional spaces, right? Because, in general, spectrum and 'set of eigenvalues' are not the same thing.
 
  • #5
heh, sorry about that. a unitary operator in a Hilbert space is what I'm working with
 
  • #6
Raven2816 said:
hmmm, i know that, so i have Ux = Lx and L is 1...so then Ux = x...? I'm just getting lost
L may not be 1, and you don't know what L is. But what is the length of Ux?
 
  • #7
Let x be an e-vector of U with e-value L as above.What do we know?

<x,x>=<Ux,Ux>

because U is unitary. If you don't see that then consider the intermediate steps:

<x,x>=<Ix,x>=<U*Ux,x>=<Ux,Ux>

Note we've just used the unitariness of U. So now we've got to use the fact that x is an e-vector

<x,x>=<Ux,Ux>=<Lx,Lx>=...?
 
  • #8
and <Lx, Lx> is the inner product of an e-vector with its e-value...so do i get one? or am i using L=1?
 
  • #9
Raven2816 said:
and <Lx, Lx> is the inner product of an e-vector with its e-value...so do i get one? or am i using L=1?
L is a number. There is a formula for pulling a number multiplier out of an inner product.
 
  • #10
a formula? isn't <Lx, Lx> = ||Lx||^2?
and i know that L<x, y> = <Lx, y> ...
 
  • #11
Raven2816 said:
i know that L<x, y> = <Lx, y> ...
What about <x,Ly>?
What about <Lx,Lx>?
 
  • #12
Raven, could you satisfy my curiosity? Are you taking a course, or reading a book on your own? What is the name and level of the course or the name of the book?
 
  • #13
ahhh i see what you mean! thanks!
 

FAQ: Unitary Operators: Why is Spectrum on Unit Circle?

What is a unitary operator?

A unitary operator is a mathematical concept used in linear algebra and functional analysis. It is a type of linear transformation that preserves the inner product between two vectors, meaning that the length and angle between the two vectors remains the same after the transformation.

What is the significance of the spectrum being on the unit circle?

The spectrum of a unitary operator refers to the set of all possible eigenvalues of the operator. When the spectrum is on the unit circle, it means that all eigenvalues have a magnitude of 1. This is significant because it implies that the operator is bijective, meaning it has both an inverse and a unique solution.

How are unitary operators related to quantum mechanics?

In quantum mechanics, unitary operators are used to describe the evolution of a quantum system over time. They represent the transformations that occur to a system as it interacts with its environment, and they preserve the probabilities of measurement outcomes.

Can unitary operators be used in other fields besides mathematics and physics?

Yes, unitary operators have applications in various fields such as signal processing, control theory, and computer science. In these fields, they are used for tasks such as filtering, error correction, and data compression.

How are unitary operators different from other types of linear transformations?

Unitary operators have the unique property of preserving the norm of a vector, meaning that the length of a vector remains the same after the transformation. Other types of linear transformations, such as orthogonal operators, may change the length of a vector. Additionally, unitary operators are always invertible, while other types of transformations may not have an inverse.

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