Results 1 to 2 of 2

Math Help - about diagonalization

  1. #1
    Newbie
    Joined
    Mar 2009
    From
    São Paulo- Brazil
    Posts
    22

    about diagonalization

    Hello,

    I want to write the following the matrix
    0 \beta 0
    1 0 0
    0 0 0

    in the form C^{-1}AC


    over Mat(3,\mathbb{Q}(\sqrt{\beta})) and A is a diagonal matrix. I know that \sqrt{\beta}\notin \mathbb{Q}.


    If there is a method to solve that kind of problem I would apperciate your help.

    Thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by Biscaim View Post
    Hello,

    I want to write the following the matrix
    0 \beta 0
    1 0 0
    0 0 0

    in the form C^{-1}AC


    over Mat(3,\mathbb{Q}(\sqrt{\beta})) and A is a diagonal matrix. I know that \sqrt{\beta}\notin \mathbb{Q}.


    If there is a method to solve that kind of problem I would apperciate your help.

    Thanks in advance.
    we really don't need the condition \sqrt{\beta} \notin \mathbb{Q}. what we need for our matrix to be diagonalizable is \beta \neq 0. let B=\begin{pmatrix}0 & \beta & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}. the eigenvalues of B are 0, \pm \sqrt{\beta}. thus: A=\begin{pmatrix}0 & 0 & 0 \\ 0 & \sqrt{\beta} & 0 \\ 0 & 0 & -\sqrt{\beta} \end{pmatrix}.

    then find the eigenvectors corresponding to each eigenvalue and put them in a matrix so that the first column is the eigenvector corresponding to the first eigenvalue on A, i.e. 0 and the

    second and the third columns are the eigenvectors corresponding to \sqrt{\beta} and -\sqrt{\beta} respectively. the result would be this matrix: X=\begin{pmatrix}0 & \sqrt{\beta} & -\sqrt{\beta} \\ 0 & 1 & 1 \\ 1 & 0 & 0 \end{pmatrix}. this matrix satisfies B=XAX^{-1}.

    the matrix C that you're looking for is just the inverse of X. you'll get: C=X^{-1}=\begin{pmatrix}0 & 0 & 1 \\ \frac{1}{2\sqrt{\beta}} & \frac{1}{2} & 0 \\ -\frac{1}{2\sqrt{\beta}} & \frac{1}{2} & 0 \end{pmatrix}. [note that the entries of all matrices are in \mathbb{Q}(\sqrt{\beta}).]
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Diagonalization
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: January 23rd 2011, 02:58 PM
  2. diagonalization
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 4th 2010, 04:06 PM
  3. Diagonalization
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: April 23rd 2010, 09:17 PM
  4. A diagonalization example
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: April 28th 2009, 08:13 AM
  5. Diagonalization
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 23rd 2009, 12:31 PM

Search Tags


/mathhelpforum @mathhelpforum