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Unitary Matrix, need to find eigen values/vectors

  1. Oct 27, 2009 #1
    1. The problem statement, all variables and given/known data

    matrix:

    1/sqrt(2) i/sqrt(2) 0

    -1/sqrt(2) i/sqrt(2) 0

    0 0 1

    Find eigen values and eigen vectors and determine if it is diagonalizable



    2. Relevant equations

    The matrix is unitary because Abar*Atranspose=I (identity matrix)




    3. The attempt at a solution

    I am having problems solving for the eigenvalues and vectors because of the imaginary numbers. What I get is:

    lambda-1/sqrt(2) i/sqrt(2) 0

    -1/sqrt(2) lambda- i/sqrt(2) 0

    0 0 lambda-1


    =(lambda-1/sqrt(2))*(lambda- i/sqrt(2))*(lambda-1)-(i/sqrt(2))*(-1/sqrt(2))*(lambda-1)

    I need help getting to the next step.
    Thanks!
     
  2. jcsd
  3. Oct 27, 2009 #2

    tiny-tim

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    Hi orbitsnerd! :smile:
    (euuugh! :yuck: have a lambda: λ and a square-root: √ :wink:)

    Look at it … (λ - 1) is obviously a factor of the determinant, so you can ignore everything except the four top-left entries:

    Code (Text):
    λ - 1/√2  i/√2
     -1/√2    λ - i/√2
    so what is the determinant of that? :smile:
     
  4. Oct 27, 2009 #3
    Awesome short cut. I now have my eigenvalues as:

    λ1=1, λ2=(1+√3)/(2√2) + [(1-√3)/(2√2)]i and λ3=(1-√3)/(2√2) + [(1+√3)/(2√2)]i

    I have issues finding the eigenvectors. I know you need to plug in the values of each λ back into the original matrix and solve for e1, e2 and e3. The imaginary number throws me off in this case.


     
  5. Oct 27, 2009 #4

    tiny-tim

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    I don't see what the problem is :confused:

    just do it the usual way. :smile:
     
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