Results 1 to 8 of 8

Math Help - Eigenvalues of Orthogonal Matrices

  1. #1
    Member
    Joined
    Apr 2008
    Posts
    123

    Eigenvalues of Orthogonal Matrices

    Show that if M is orthogonal, then M admits at most one eigenvalue, and as such must be either +1 OR -1.

    Using the definition of orthogonality and eigenvalues, it's easy enough to show that the eigenvalues of M are +/- 1. I'm however unable to show that M admits at most one eigenvalue.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by h2osprey View Post
    Show that if M is orthogonal, then M admits at most one eigenvalue, and as such must be either +1 OR -1.

    Using the definition of orthogonality and eigenvalues, it's easy enough to show that the eigenvalues of M are +/- 1. I'm however unable to show that M admits at most one eigenvalue.

    This is false: M=\left(\begin{array}{rrr}1&0&1\\\!\!0&\!\!-1&\!\!0\\0&0&1\end{array}\right) is orthogonal and its characteristic polynomial is p_M(x)=(x-1)^2(x+1)\,\Longrightarrow both 1, -1 are eigenvalues of M.

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,697
    Thanks
    1469
    No, Tonio. M is not orthogonal. Two equivalent definitions of "orthogonal matrix" are:
    1) It's columns are orthogonal vectors.
    That is not true for your matrix, M, because the dot products of the first and third columns is 1, not 0.

    2) The transpose equals the inverse matrix.
    That is not true for your matrix, M, because
    \begin{pmatrix}1 & 0 & 0 \\0 & -1 & 0 \\ 1 & 0 & 1\end{pmatrix}\begin{pmatrix}1 & 0 & 1 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{pmatrix}= \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 2\end{pmatrix}.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Nov 2008
    From
    Paris
    Posts
    354
    I guess the 1 in the top right corner is a 0, just a little misprint, no?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by clic-clac View Post
    I guess the 1 in the top right corner is a 0, just a little misprint, no?

    Indeed...good sight! Thanx

    Tonio
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by HallsofIvy View Post
    No, Tonio. M is not orthogonal. Two equivalent definitions of "orthogonal matrix" are:
    1) It's columns are orthogonal vectors.
    That is not true for your matrix, M, because the dot products of the first and third columns is 1, not 0.

    2) The transpose equals the inverse matrix.
    That is not true for your matrix, M, because
    \begin{pmatrix}1 & 0 & 0 \\0 & -1 & 0 \\ 1 & 0 & 1\end{pmatrix}\begin{pmatrix}1 & 0 & 1 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{pmatrix}= \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 2\end{pmatrix}.

    Typo: the 1-3 entry should be 0. Thanx

    Tonio
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    Quote Originally Posted by h2osprey View Post
    Show that if M is orthogonal, then M admits at most one eigenvalue, and as such must be either +1 OR -1.

    Using the definition of orthogonality and eigenvalues, it's easy enough to show that the eigenvalues of M are +/- 1. I'm however unable to show that M admits at most one eigenvalue.
    What about M=\begin{pmatrix}-1&0\\0&1\end{pmatrix}?

    What about the matrix of a rotation of angle \theta? It has eigenvalues e^{i\theta},e^{-i\theta}. But perhaps you were dealing with real eigenvalues?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member
    Joined
    Apr 2008
    Posts
    123
    Thanks for all the help! It didn't ever occur to me that the question could be wrong, but obviously the counterexample was simple enough.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: August 15th 2011, 04:32 AM
  2. Replies: 3
    Last Post: May 10th 2011, 01:01 AM
  3. Orthogonal matrices
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 9th 2010, 10:00 PM
  4. Eigenvalues of orthogonal matrices
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 28th 2010, 01:43 AM
  5. Eigenvalues, eigenvectors, finding an orthogonal matrix
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: May 5th 2009, 11:06 AM

Search Tags


/mathhelpforum @mathhelpforum