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Unitary Matrix preserves the norm Proof

  1. Nov 4, 2015 #1

    RJLiberator

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    Gold Member

    1. The problem statement, all variables and given/known data
    Let |v> ∈ ℂ^2 and |w> = A|v> where A is an nxn unitary matrix. Show that <v|v> = <w|w>.

    2. Relevant equations
    * = complex conjugate
    † = hermitian conjugate

    3. The attempt at a solution

    Start: <v|v> = <w|w>
    Use definition of w
    <v|v>=<A|v>A|v>>

    Here's the interesting part
    Using properties of a complex inner product, we can take out the unitary matrix A on the right hand side. When we do so, what do we get?

    Is it A*A or is it A†A ?

    If it is A†A this proof is complete, as sine A is unitary, this means A†A = the identity matrix and we get equality.

    If the definition is A*A then we have to do more work.

    I am getting mixed up, as I'm seeing both definitions floating around and with the abusive notation everywhere it is making me second guess which definition is right.

    Is there any clarity on this issue here?
     
  2. jcsd
  3. Nov 4, 2015 #2

    fresh_42

    Staff: Mentor

  4. Nov 4, 2015 #3

    RJLiberator

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    Thanks for the links. I was aware of the Unitary link, but the other non-English wiki link is of interest.

    Although, the language barrier makes it difficult to follow.

    They show that
    <Ax, Ay> = <x,A^HAy> = <x,A^-1Ay> = <x, Iy> = <x,y>

    Kind of similar to what I am doing, I guess, but I'm not sure what "H" is, I am guessing Hermitian conjugate.
    So they take the Hermitian conjugate, but then get an inverse somehow, and A^-1*A is clearly the identity.

    Still kind of unclear :/
     
  5. Nov 5, 2015 #4

    Samy_A

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    In the German Wikipedia, ##A^H## is indeed the Hermitian conjugate of ##A##, and then they use the fact that A is a Unitary matrix.
     
  6. Nov 5, 2015 #5

    RJLiberator

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    So this is a definition of complex inner product =

    <Ax,Ay> = <x,A^HAy>

    If so, then this is rather easy as A^HA = Identity matrix by definition of Unitary.
     
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