Results 1 to 5 of 5

Math Help - Diagonalisation of symmetric matricies

  1. #1
    Newbie
    Joined
    May 2011
    Posts
    9

    Diagonalisation of symmetric matricies

    I have just learnt how to reduce a symmetric matrix to a diagonal one, using the equation below:

    D={P}^{T}AP
    Where A is the original symmetric matrix and D is the diagonal matrix.
    P is an orthogonal matrix whose columns consist of the normalised eigenvectors of A.

    My problem is how to write the above equation in terms of A.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by incblue View Post
    I have just learnt how to reduce a symmetric matrix to a diagonal one, using the equation below:

    D={P}^{T}AP
    Where A is the original symmetric matrix and D is the diagonal matrix.
    P is an orthogonal matrix whose columns consist of the normalised eigenvectors of A.

    My problem is how to write the above equation in terms of A.
    You just need to solve for A.

    First multiply on the left by P and remember that PP^T=P^TP=I

    PD=PP^TAP=AP Now multiply on the right by P^T

    PDP^T=APP^T=A \iff A=PDP^T
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    May 2011
    Posts
    9
    Quote Originally Posted by TheEmptySet View Post
    Now multiply on the right by P^T

    PDP^T=APP^T=A \iff A=PDP^T
    I am struggling with this step. I thought that you always had to multiply on the left.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by incblue View Post
    I am struggling with this step. I thought that you always had to multiply on the left.
    You can multiply on either side but remember that matrix multiplication does not commute.

    For example in the real numbers is we have

    \frac{1}{2}x=5 if we multiply on the left or the right by 2 we can still solve for x by the communicative property, but with matrices it is not so nice.

    A\mathbf{x}=\mathbf{b} If we multiply on the left by the inverse of A we get

    A^{-1}A\mathbf{x}=A^{-1}\mathbf{b} \iff \mathbf{x} =A^{-1}\mathbf{b}

    but if you multiply on the right you get

    A\mathbf{x}{A^{-1}}=\mathbf{b}A^{-1}

    and since multiplication does not commute we cannot simplify.
    Last edited by TheEmptySet; June 5th 2011 at 10:36 AM. Reason: LaTex error
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    May 2011
    Posts
    9
    Okay, thanks for the explanation.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: April 5th 2013, 08:23 PM
  2. Replies: 3
    Last Post: March 17th 2011, 05:34 AM
  3. Replies: 22
    Last Post: January 12th 2011, 03:52 AM
  4. Symmetric relation v.s. symmetric matrix
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 14th 2010, 11:37 PM
  5. Diagonalisation of a matrix
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: December 25th 2007, 06:52 AM

Search Tags


/mathhelpforum @mathhelpforum