Results 1 to 2 of 2

Math Help - Diagonalizable matrix

  1. #1
    Newbie
    Joined
    Jun 2009
    Posts
    5

    Diagonalizable matrix

    Show that a diagonalizable matrix having only one eigenvalue is a scalar matrix.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Random Variable's Avatar
    Joined
    May 2009
    Posts
    959
    Thanks
    3
    Let A be a diagonalizable matrix having only one eigenvalue  \lambda .

    By the defintion of diagonalizability, there exists an nonsingular matrix P such that  B = P^{-1}AP (where B is a diagonal matrix).

    The diagonal entries of B are the eigenvalues of A; therefore,  B = \lambda I (which is a scalar matrix).

    This means that A is similar to  \lambda I . But the only matrix similar to a scalar matrix is itself. So A is a scalar matrix.

    I coincidentally just proved the last part in another thread about 20 minutes ago.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. How to tell if a matrix is diagonalizable?
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: May 14th 2011, 07:41 PM
  2. Diagonalizable Matrix
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 20th 2010, 03:38 AM
  3. Diagonalizable matrix
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 5th 2010, 04:32 PM
  4. Replies: 0
    Last Post: March 20th 2008, 09:27 AM
  5. Diagonalizable Matrix
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 14th 2006, 05:39 PM

Search Tags


/mathhelpforum @mathhelpforum