1. Not finding help here? Sign up for a free 30min 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!

Orthogonal Matrix

  1. Dec 10, 2009 #1
    1. The problem statement, all variables and given/known data

    Let l be an eigenvalue of an orthogonal matrix A, where l = r + is. Prove that l * conj(l) = r^2 + s^2 = 1.

    2. Relevant equations



    3. The attempt at a solution

    I am really confused on where to go with this one.

    I have Ax = A I x = A A^T A x = l^3 x

    and Ax = l x so l x = l^3 x

    l = l^3

    l^2 = 1
    l = 1 or -1

    But I can't really figure out what to do from here, am I even on the right track?

    Thanks for the help
     
  2. jcsd
  3. Dec 10, 2009 #2
    Hi, i am just studying linear algebra (the final is coming next week, so stressed!!).

    I think the question about l*conjugate(l) = r^2+s^2 = 1 just mean the length of l is 1

    But i think if Matrix with complex entries, say M, is orthogonal means M is unitary. So the length of every eigenvalue is 1

    since unitary matrix won't change vector's langth so its eigenvalues' length is always 1
     
  4. Dec 10, 2009 #3

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    phrygian cannot use this fact; he/she has to prove it.

    phrygian, the cube route isn't really going to help here. You went one step too far.

    You know that [tex]\mathbf A \vec x = \lambda \vec x[/tex]. The conjugate transpose of the right-hand side is [tex](\lambda \vec x)^* = \vec x^*\lambda^*[/tex]. What is the matrix product of this conjugate transpose with [tex] \lambda \vec x[/tex]?
     
  5. Dec 11, 2009 #4
    Is there a way to do this without using conjugate transposes? The book has a hint that says first show that ||Ax|| = ||x|| for any vector x, but i just cant seem to get past that first step
     
  6. Dec 12, 2009 #5

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    What is the precise definition of "orthogonal" matrix? Some texts use that a matrix, Q, is orthogonal if and only if <Qu, v>= <u, Qv> for all vector u and v (< u, v> is the inner product). From that it is not too hard to show that ||Qv||= ||v||.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Orthogonal Matrix
  1. Orthogonal Matrix (Replies: 4)

  2. Orthogonal matrix (Replies: 15)

  3. Orthogonal MAtrix (Replies: 4)

  4. Orthogonal Matrix (Replies: 8)

  5. Orthogonal Matrix (Replies: 5)

Loading...