Results 1 to 4 of 4

Math Help - commuting matrices and simultaneous diagonalization

  1. #1
    Newbie
    Joined
    Sep 2010
    Posts
    8

    commuting matrices and simultaneous diagonalization

    Hi,

    I'm training for exam and come across a question from one of the exams:

    T and S are normal linear transformations. Also they commute TS = ST.
    It is required to prove that there exists common vector basis consisting of T's (or S's) eigenvectors.
    One way to prove this is to find a matrix U such that:

    U^{-1}[T]U = diag(\gamma_i) \mbox{ and } U^{-1}[S]U = diag(\lambda_i)

    After some manipulation I can show that TS and ST are also normal, not sure if this is useful.
    I know (i searched the web to find more clues) that commute matrices should have common eigenvector, though again i didn't find simple proof for this fact.

    Can anyone please give some direction of how to prove this ?

    Thank you

    P.S.
    Our material doesn't include things like classification into groups/subgroups or representation theory so proof should use more elementary concepts like eigenvector, eigenvalues, unitary matrices
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    Re: commuting matrices and simultaneous diagonalization

    This is a very common theorem. See here, for example.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2010
    Posts
    8

    Thumbs up Re: commuting matrices and simultaneous diagonalization

    Thanks for the suggesting the site.
    I already seen that proof and I find it a bit wordy...
    I'm thinking why to involve scalar matrices and why the need to induct on vector space dimension... (I understand that this might be also a way to prove)

    Eventually I find a very nice and clear proof here: on page 8.
    Thanks
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    Re: commuting matrices and simultaneous diagonalization

    Quote Originally Posted by ktotam View Post
    Thanks for the suggesting the site.
    I already seen that proof and I find it a bit wordy...
    I'm thinking why to involve scalar matrices and why the need to induct on vector space dimension... (I understand that this might be also a way to prove)

    Eventually I find a very nice and clear proof here: on page 8.
    Thanks
    Never forget Keith Conrad. He usually has uncannily clear explanations of things. I'm glad you're happy!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Finding commuting matrices
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: April 2nd 2013, 09:37 AM
  2. [SOLVED] theorem about commuting matrices.
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: August 12th 2011, 09:14 AM
  3. Commuting Matrices
    Posted in the Math Challenge Problems Forum
    Replies: 3
    Last Post: August 20th 2010, 08:15 AM
  4. Spectral radius of sum of commuting matrices
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: July 3rd 2010, 08:47 PM
  5. simultaneous diagonalization
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: December 20th 2009, 11:13 PM

Search Tags


/mathhelpforum @mathhelpforum