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Math Help - Unitary Matrices

  1. #1
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    Unitary Matrices

    The question:
    Show that ||UX|| = ||X||.

    My attempt


    Say X = [x1 x2 ... xn]

    Now I know that ||X||2 = <X, X> = |x1|2 ... |xn|2

    I'm not sure where to go from here. Can anyone help?
    Last edited by apple123; March 29th 2010 at 07:44 PM.
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  2. #2
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    Quote Originally Posted by apple123 View Post
    The question:
    U is a unitary matrix. Show that
    ||UX|| = ||X|| for all X in the complex set.

    Also show that |λ| = 1 for every eigenvalue λ of U.

    My attempt
    I'm not sure where to start. So I looked up the definition of a unitary matrix.


    You had to? I mean, if you're asked to do this exercise then one could expect you know at least what a unitary matrix is


    It satisfies one of these conditions:
    U-1 = UH
    The rows of U are an orthonormal set in the complex set
    The columns of U are an orthonormal set in the complex set


    Who knows what you meant by the first condition. Anyway, a matrix U is unitary if U^{-1}=U^{*}=\overline{U^t}\Longleftrightarrow U^{*}U=UU^{*}=I , and then

    we have \|x\|^2=<x,x>=<x,U^{*}Ux>=<Ux,Ux>=\|Ux\|^2 and we're done


    Say X = [x1 x2 ... xn]

    Now I know that ||X||2 = <X, X> = |x1|2 ... |xn|2


    This should be x=(x_1,\ldots,x_n)\Longrightarrow \|x\|^2=\sum^n_{i=1}|x_i|^2 , and not a product.

    Tonio


    I'm not sure where to go from here. Can anyone help?
    .
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