Results 1 to 4 of 4

Thread: Similarity and Digonalization in matrices

  1. #1
    Junior Member
    Joined
    Nov 2006
    Posts
    48

    Similarity and Digonalization in matrices

    a) If A and B are invertible matrices show that AB and BA are similar

    b) Let A be an invertible matrix. Prove that if A is digonalizable, so is its inverse.(A^-1)

    Thank You
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    10
    Quote Originally Posted by somestudent2 View Post
    a) If A and B are invertible matrices show that AB and BA are similar
    BA = A^{-1}(AB)A.

    Quote Originally Posted by somestudent2 View Post
    b) Let A be an invertible matrix. Prove that if A is digonalizable, so is its inverse.(A^-1)
    If A is diagonalisable then P^{-1}AP = D. What happens when you take the inverse of both sides of that equation?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Nov 2006
    Posts
    48
    Hi

    for the second one by taking inverse on both sides we arive at

    P^-1A^-1P = D^-1 ?

    And this proves it (does it matter that Diagonal matrix is inversed now)? Or I need to make some additional calculations?

    Thanks
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    10
    Quote Originally Posted by somestudent2 View Post
    for the second one by taking inverse on both sides we arive at

    P^-1A^-1P = D^-1 ?

    And this proves it (does it matter that Diagonal matrix is inversed now)? Or I need to make some additional calculations?
    If a diagonal matrix has an inverse then the inverse is also diagonal. So all you need to do is to explain why D is invertible. (Reason: the diagonal elements of D are the eigenvalues of A and, since A is invertible, none of them can be zero.)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Similarity Matrices
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Dec 4th 2010, 03:30 AM
  2. Need help with determining Similarity of two matrices
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: Jul 26th 2010, 04:31 AM
  3. Similarity of matrices...
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 20th 2010, 04:35 AM
  4. classify up to similarity (matrices)?? meaning?
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Mar 16th 2010, 03:53 AM
  5. Similarity of Matrices
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: Nov 5th 2009, 10:14 AM

Search tags for this page

Click on a term to search for related topics.

Search Tags


/mathhelpforum @mathhelpforum