Results 1 to 4 of 4

Math Help - Matrix eigenvalues

  1. #1
    Newbie
    Joined
    Jan 2009
    Posts
    3

    Matrix eigenvalues

    If you are told that a square matrix M is diagonalisable, and it has only one eigenvalue which is c, then how do you show that M=cI, where I is the identity matrix?

    Alternatively, and more generally, can you say that the sum of the eigenvalues of a diagonalisable square matrix M is the sum of the values along the main diagonal of M? How would you prove that?
    Last edited by GrassyRoots; January 28th 2009 at 05:28 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jan 2009
    Posts
    3
    I think I understand the second part. What about the 1st part?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6
    Quote Originally Posted by GrassyRoots View Post
    I think I understand the second part. What about the 1st part?
    Proof: Applying the definition of diagonalizability: M is diagble \implies \exists invertible P such that P^{-1} M P =\Sigma. Since all its eigen values are c, \Sigma = cI. But P^{-1} M P = cI \implies M = P(cI) P^{-1} = cI. Thus \Sigma = cI \implies M = cI \, \quad \blacksquare
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: November 27th 2010, 04:07 PM
  2. eigenvalues - 3x3 matrix
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: December 6th 2009, 02:04 PM
  3. Eigenvalues of a 3x3 matrix
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: May 4th 2009, 10:44 PM
  4. Eigenvalues of a 3*3 Matrix
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: November 7th 2008, 03:10 AM
  5. 3x3 Matrix - Eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 10
    Last Post: October 2nd 2008, 11:06 AM

Search Tags


/mathhelpforum @mathhelpforum