Results 1 to 3 of 3

Math Help - Properties of Commutative Matrix

  1. #1
    Newbie
    Joined
    Jul 2008
    Posts
    5

    Properties of Commutative Matrix

    If A and C are both nxn matrices that have n distinct different eigenvalues.

    AC=CA.

    How do you show that the eigenvectors of A and C are the same? How do the eigenvalues relate?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    Posts
    461
    Quote Originally Posted by ebot View Post
    If A and C are both nxn matrices that have n distinct different eigenvalues.

    AC=CA.

    How do you show that the eigenvectors of A and C are the same? How do the eigenvalues relate?
    I'm not sure about this:

    I is Identity matrix

     det(AC - t I) = det ( A(C-A^{-1}t))

     = det(A) * det((C-A^{-1}t))

     = det(C-A^{-1}t) * det(A)

     = det ( (C-A^{-1}t)*A )

     = det(CA - A^{-1}A t)

     = det(CA - I*t)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by ebot View Post
    If A and C are both nxn matrices that have n distinct different eigenvalues and AC=CA, how do you show that the eigenvectors of A and C are the same? How do the eigenvalues relate?
    suppose v is an eigenvector of A. so Av=\lambda v, for some scalar \lambda. let V_{\lambda} be the eigenspace corresponding to \lambda. since all eigenvalues of A are distinct, V_{\lambda} is one dimensional, i.e. V_{\lambda}=<v>.

    now we have \lambda Cv=CAv=ACv. thus Cv \in V_{\lambda}=<v>. hence Cv=\mu v, for some scalar \mu. that means v is also an eigenvector of C. so we've proved that every eigenvector of A is an

    eigenvector of C. a similar argument shows that every eigenvector of C is an eigenvector of A. \ \Box
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Commutative and Associative properties.
    Posted in the Advanced Algebra Forum
    Replies: 10
    Last Post: June 5th 2012, 05:30 PM
  2. Replies: 1
    Last Post: April 26th 2010, 08:18 AM
  3. Commutative matrix
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 17th 2009, 01:24 PM
  4. Exercise in Matrix Properties
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: July 28th 2009, 03:56 PM
  5. commutative matrix
    Posted in the Algebra Forum
    Replies: 1
    Last Post: January 6th 2009, 07:32 AM

Search Tags


/mathhelpforum @mathhelpforum