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Transition matrix and rational canonical form

  1. Aug 22, 2013 #1
    1. The problem statement, all variables and given/known data

    I want to find the transition matrix for the rational canonical form of the matrix A below.

    2. Relevant equations



    3. The attempt at a solution

    Let ##A## be the 3x3 matrix

    ##\begin{bmatrix} 3 & 4 & 0 \\-1 & -3 & -2 \\ 1 & 2 & 1 \end{bmatrix}##


    The characterisitc and minimal polynomials are both ##(x-1)^2(x+1)##

    The eigenspace for 1 is
    ##\{ \begin{bmatrix} 2 \\-1 \\ 1 \end{bmatrix} \}##


    The eigenspace for -1 is:
    ##\{ \begin{bmatrix} 2 \\-2 \\ 1 \end{bmatrix} \}##


    The rational canonical form ##R## is:

    ##\begin{bmatrix} -1 & 0 & 0 \\0 & 0 & -1 \\ 0 & 1 & 2 \end{bmatrix}##



    I want to find the transition matrix ##P## such that ##A=PRP^{-1}##

    I thought we had to find 3 independent vectors...one from the eignspace of 1, another from the eigenspace of -1, and then any other third vector such that the three would be linearly independent. So I chose P to be:

    ##\begin{bmatrix} 2 & 2 & 1 \\-2 & -1 & 0 \\ 1 & 1 & 0 \end{bmatrix}##



    But when I multiplied ##PRP^{-1}##, I did not get ##A##...I'm not sure why.

    I would appreciate it if anybody could tell me where I went wrong and how I can fix it.

    Thanks in advance
     
  2. jcsd
  3. Aug 23, 2013 #2
    Hi!

    edit: sorry, I misread
     
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