4
$\begingroup$

We are asked to find an invertible matrix $P$ and an upper triangular matrix $U$ such that:

$P^{-1}\begin{pmatrix} 3 & -1 & 1 \\ 2 & 0 & 0 \\ -1 & 1 & 3 \end{pmatrix}P=U$

I'm a bit stuck.

I found the characteristic polynomial of the matrix, it is $(-x+2)^3$, so the only eigenvalue is 2.

The problem comes with the eigenvectors. I could only find one, $\begin{pmatrix} 1 \\ 1\\0 \end{pmatrix}$ ...How do I go on from here? the zero vectors are not eigenvectors, and even if they were,

the matrix $\begin{pmatrix} 1 & 0 & 0\\ 1 & 0 & 0\\0 & 0 & 0 \end{pmatrix}$ is singular so it can't be the $P$ i'm looking for...

$\endgroup$
  • $\begingroup$ From Jordan normal form, we have $$ \begin{bmatrix} 3 & -1 & 1\\ 2 & 0 & 0\\ -1 & 1 & 3 \end{bmatrix} \sim \begin{bmatrix} 2 & 1 & 0\\ 0 & 2 & 1\\ 0 & 0 & 2 \end{bmatrix} $$ $\endgroup$ – user17762 Nov 30 '13 at 20:11
  • 1
    $\begingroup$ you need to look for generalized eigenvectors of the form $(A-\lambda I)^k v =0$, and then the matrix $U$ will be a Jordan matrix. $\endgroup$ – Slugger Nov 30 '13 at 20:11
  • $\begingroup$ Slugger I've never done that, this is the first time i encounter such a question and I'm not sure what you mean. Could you elaborate / demonstrate? $\endgroup$ – Oria Gruber Nov 30 '13 at 20:17
  • $\begingroup$ Well, as you said there is only one eigenvalue and only one eigenvector. When this is the case we know that we can factor the original matrix (call it $A$) in the form $A = PUP^{-1}$ where $U$ is a jordan matrix, it is a matrix with eigenvalue on the diagonal and ones just above (see comment by user17762). The $P$ matrices are found using eigenvectors and generalized eigenvectors. Generalized eigenvectors are found by $(A-\lambda I)^2v=0$, $(A-\lambda I)^3v=0$ etcetera. see en.wikipedia.org/wiki/Jordan_normal_form $\endgroup$ – Slugger Nov 30 '13 at 20:51
1
$\begingroup$

If you don't know about JNF, here's another process which is easily generalizable. What I'm doing is simply following the constructive proof of Schur's Decomposition, (link provided below). Every now and then I'll choose random vectors that satisfy certain properties. There is an infinite number of choices for these vectors. You should pick your own while mimicing my answer. The $U$ and $P$ you'll end up with will probably be different.

Let $A=\begin{bmatrix} 3 & -1 & 1 \\ 2 & 0 & 0 \\ -1 & 1 & 3 \end{bmatrix}$.

You got only one eigenvector, namely $v_1:=\begin{bmatrix} 1 & 1 & 0\end{bmatrix}^T$.

Consider $P_1:=\begin{bmatrix} v_1 \mid v_2 \mid v_3\end{bmatrix}$ by columns. You want an invertible $P$ so just let $v_2, v_3$ be such that $P_1$ is invertible. An easy choice is $v_2:=\begin{bmatrix} 1 & -1 & 0\end{bmatrix}^T$ and $v_3:=\begin{bmatrix} 0 & 0 & 1\end{bmatrix}^T$. It's easy to see $P$ is invertible because its columns are orthogonal.

This yields $P_1^{-1}AP_1=\begin{bmatrix} 2 & 3 & 1/2 \\ 0 & 1 & 1/2 \\ 0 & -2 & 3 \end{bmatrix}$. It's not an upper triangular matrix. Let $B=P_1^{-1}AP_1$.

Suppose for a moment that there are matrices $P_2$ (invertible) and $T$ such that $P_2^{-1}BP_2=T$ where $T$ is an upper triangular matrix. This would yield $B=P_2TP_2^{-1}$ and $P_2TP_2^{-1}=P_1^{-1}AP_1$, thus giving $T=(P_2^{-1}P_1^{-1})A(P_1P_2)$.

So let's (try to) triangularize $B$.

Repeating the process wouldn't help, so let's instead try to triangularize $\color{grey}{B_1:=}\begin{bmatrix} 1 & 1/2 \\ -2 & 3\end{bmatrix}$. (Why? See Schur decomposition theorem's proof by induction here (page 12) ).

It's easy to check that $\left(2, \begin{bmatrix} 1\\ 2\end{bmatrix}\right)$ is an eigenpair of $B_1$ and that $B_1$ doesn't have any other linearly independent eigenvectors.

Define $P_{B_1}:=\begin{bmatrix}1 & -2\\2 & 1\end{bmatrix}$. The second column was chosen just to make $P_{B_1}$ invertible. There are, of course, other possibilities.

Then $P_{B_1}^{-1}B_1P_{B_1}=\begin{bmatrix}2 & 5/2\\ 0 & 2 \end{bmatrix}$.

Now it's possible to construct the aforementioned $P_2$. Let $P_2=\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & -2\\ 0 & 2 & 1\end{bmatrix}$.

Block multiplication assures $P_2$ does the job.

Indeed $P_2^{-1}P_1^{-1}AP_1P_2=P_2^{-1}BP_2=\begin{bmatrix} 2 & 4 & -11/2\\ 0 & 2 & 5/2\\ 0 & 0 & 2\end{bmatrix}$.

So just let $U:=\begin{bmatrix} 2 & 4 & -11/2\\ 0 & 2 & 5/2\\ 0 & 0 & 2\end{bmatrix}$ and $P:=P_1P_2\color{grey}{=\begin{bmatrix} 1 & 1 & -2\\ 1 & -1 & 2\\ 0 & 2 & 1\end{bmatrix}}$.

Let's confirm it works. First find $P^{-1}=\begin{bmatrix}1/2 & 1/2 & 0\\ 1/10 & -1/10 & 2/5\\ -1/5& 1/5 & 1/5 \end{bmatrix}$.

Then $$\begin{align} P^{-1}AP&=\begin{bmatrix}1/2 & 1/2 & 0\\ 1/10 & -1/10 & 2/5\\ -1/5& 1/5 & 1/5 \end{bmatrix}\begin{bmatrix} 3 & -1 & 1 \\ 2 & 0 & 0 \\ -1 & 1 & 3 \end{bmatrix}\begin{bmatrix} 1 & 1 & -2\\ 1 & -1 & 2\\ 0 & 2 & 1\end{bmatrix}\\ &=\begin{bmatrix} 5/2 & -1/2 & 1/2\\-3/10 & 3/10 & 13/10\\ -2/5 & 2/5 & 2/5\end{bmatrix}\begin{bmatrix} 1 & 1 & -2\\ 1 & -1 & 2\\ 0 & 2 & 1\end{bmatrix}\\ &=\begin{bmatrix} 2 & 4 & -11/2\\ 0 & 2 & 5/2\\ 0 & 0 & 2\end{bmatrix}.\end{align}$$

$\endgroup$
0
$\begingroup$

You can take for $P$ take the following matrix: $$ P=\begin{pmatrix}-2 & 1 & 1\cr -2 & 2 & 0\cr 0 & -1 & 0 \end{pmatrix} $$ Then $P^{-1}AP$ is upper-triangular, and in canonical Jordan form. Note that $P$ has integer entries, and $P^{-1}$ has rational entries. You can find such a matrix by direct calculation as a system of equations in the entries of $P$, or by computing the bases of the generalised eigenspaces.

$\endgroup$
  • 1
    $\begingroup$ How did you find this P? $\endgroup$ – Oria Gruber Nov 30 '13 at 20:18

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.