Proof of an upper triangular matrix - theory

[PirateFan308]

Homework Statement

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

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

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!

 

[mighty2000]

Thank you for your post. It seems like you have already made good progress in proving the linear independence of (A-λI)^2v, (A-λI)v, and v. To prove the matrix of Y^{-1}AY, we can use the fact that Y is an invertible matrix, so we can write Y^{-1}AY=A. We can then use the given information that (A-λI)^3=0 to simplify our calculations. Let's start by expanding (A-λI)^2v:

(A-λI)^2v = (A-λI)(A-λI)v = (A-λI)v_2

where v_2=(A-λI)v. Now, using the hint provided by your professor, we know that (A-λI)v_2=v_1, where v_1=(A-λI)^2v. This means that (A-λI)^2v=v_1=v_1-v_2. We can then rewrite Y^{-1}AY as:

Y^{-1}AY = Y^{-1}(A-λI)^2v + Y^{-1}(A-λI)v + Y^{-1}v

= Y^{-1}(v_1-v_2) + Y^{-1}v_2 + Y^{-1}v

= Y^{-1}v_1

Now, since we are given that v_1 is not in Ker(A-λI)^2, we know that Y^{-1}v_1 is a nonzero vector. Therefore, we can use this vector to construct the matrix of Y^{-1}AY:

Y^{-1}AY = \begin{pmatrix}Y^{-1}v_1 & 0 & 0\\0 & Y^{-1}v_1 & 0\\0 & 0 & Y^{-1}v_1\end{pmatrix}

Finally, if we substitute λ=3 into this matrix, we get:

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

which is the desired result. I hope this helps! Let me know if you have any further questions.