1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Eigenvalue proof help

  1. Jan 12, 2008 #1
    I know that if T has eigenvalue k, then T* has eigenvalue k bar. But if T has eigenvector x, does T* also have eigenvector x? If so, how do you prove it? I don't see that in my textbook.
  2. jcsd
  3. Jan 12, 2008 #2


    User Avatar
    Science Advisor

    There is no proof because it is not true. For example, if
    [tex]T= \left[\begin{array}{cc}0 & i \\i & 0\end{array}right][/tex]
    Then T has eigenvalue i, with eigenvecor [a, a] and eigenvalue -i with eigenvector [1, -1].
    [tex]T*= \left[\begin{array}{cc}0 & -i \\ -i & 0\end{array}\right][/tex]
    which also eigenvalue i but now with eigenvectors [a, -a] and eigenvalue -i with eigenvectors [a, a].
  4. Jan 12, 2008 #3
    Ok, I just read that it is true if T is a normal operator. Thanks.
  5. Jan 12, 2008 #4


    User Avatar
    Science Advisor
    Homework Helper

    What about the converse? Namely, if T and T* share their eigenvectors, is T necessarily normal?
  6. Jan 12, 2008 #5
    I don't know if you are asking rhetorically or not. But my textbook doesn't state the converse.
  7. Jan 12, 2008 #6


    User Avatar
    Science Advisor
    Homework Helper

    I'm just throwing it out there. It may be a good exercise to think about this.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Eigenvalue proof help
  1. Eigenvalue proof (Replies: 1)

  2. Eigenvalue proof (Replies: 23)

  3. Eigenvalue Proof (Replies: 5)

  4. Eigenvalues proof (Replies: 7)

  5. Eigenvalue proof (Replies: 6)