# Proof of an upper triangular matrix - theory

1. Jan 24, 2012

### PirateFan308

1. The problem statement, all variables and given/known data
Suppose that A is a 3x3 matrix, having characteristic polynomial $(λ-x)^{3}$. Suppose that there is a vector $v$ in $F^3$ which is not in $Ker(A-λI)^2$. You may assume that $(A-λI)^3=0$. Show that $(A-λI)^2v, (A-λI)v, v$ are linearly independent. If Y is the matrix having columns $(A-λI)^2v, (A-λI)v, v$ show that if λ=3 then

$$Y^{-1}AY = \begin{pmatrix}3 & 1 & 0\\0 & 3 & 1\\0 & 0 & 3\end{pmatrix}$$

3. The attempt at a solution
I have already proved that $(A-λI)^2v, (A-λI)v, v$ are linearly independent by using the fact that $(A-λI)^3=0$ and $(A-λI)^2≠0$. I have no idea where to start in proving the matrix of $Y^{-1}AY$.

The hints our prof gave us was that we want:
$Av_1=3(A-3I)^2v = A(A-3I)^2v$

$(A-3I)^3v=0 ~so~ (A-3I)(A-3I)^2v=0~~=>~~(A-3I)v_1=0$

For $v_2=(A-3I)v ~~~~ (A-3I)v_2=v_1$

But I'm not quite sure how this helps. Thanks!

Last edited by a moderator: Jan 24, 2012