Transition matrix and rational canonical form

[Artusartos]

Homework Statement

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

Homework Equations

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

 

[saxen]

Hi!

edit: sorry, I misread