Find Jordan canonical form and basis of a linear operator.

Question

Let $T:\mathbb{R}^3 \to \mathbb{R}^3$ be a linear operator such that: $T(x,y,z)=(-y-2z,x+3y+z,x+3z)$, I need to find a Jordan canonical form and a basis. This is what i did:

In the first place, I found the associated matrix to this linear operator in the canonical basis which is this one:

$$A=\begin{pmatrix} 0 & -1 & -2 \\ 1 & 3 & 1 \\ 1 & 0 & 3 \\ \end{pmatrix}$$ After that i found the characteristic polynomial which is: $(\lambda-2)^3=0$ so we have this polynomial that has only one root with multiplicity 3.

After finding the eigenvalue I found the eigenvector associated to 2, which is $V_3=(-1,0,1)$.

Now the Jordan canonical form should be this one(If I have done it correctly):

$$J=\begin{pmatrix} 2 & 0 & 0 \\ 1 & 2 & 0 \\ 0 & 1 & 2 \\ \end{pmatrix}$$

We know the a Jordan basis is formed with these vectors: $B=v_1,v_2,v_3$

First I found $v_2$ :

$A.v_2=2.v_2+1.v_3$, which is equal to : $$ \begin{align*} \begin{cases} 2x+y+2z &=1\\ x+y+z&=0\\ x+z&=1 \end{cases} \end{align*} $$So $v_2=(-1,-1,1)$

The problem is here with the last vector: After doing the same operation I get to this point: $A.v_1=2.v_1+1.v_2$, which is equal to: $$ \begin{align*} \begin{cases} 2x+y+2z &=1\\ x+y+z&=0\\ x+z&=1 \end{cases} \end{align*} $$ and clearly this system does not have solution... what am I doing wrong?

Answer 1

Usually the Jordan normal form is as follow

$$J=\begin{pmatrix} 2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \\ \end{pmatrix}$$

By Jordan theorem we know that a matrix $P$ exists such that $$P^{-1}AP=J$$

let $$P=[v_1,v_2,v_3,v_4]$$

then P has to satisfy the following system: $$AP=PJ$$ that is in this case $$Av_1=2v_1\implies (A-2I)v_1=0$$ $$Av_2=v_1+2v_2\implies (A-2I)v_2=v_1$$ $$Av_3=v_2+2v_3\implies (A-2I)v_3=v_2$$

Once we have $v_1$ we can find $v_2$ and finally $v_3$, that is

$$(A-2I)v_1=0 \implies\begin{pmatrix} -2 & -1 & -2 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \\ \end{pmatrix}v_1=0 \implies v_1=(1,0,-1)$$

$$(A-2I)v_2=v_1 \implies\begin{pmatrix} -2 & -1 & -2 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \\ \end{pmatrix}v_2=v_1 \implies v_2=(-1,1,0)$$

$$(A-2I)v_3=v_2 \implies\begin{pmatrix} -2 & -1 & -2 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \\ \end{pmatrix}v_3=v_2 \implies v_3=(0,1,0)$$

Answer 2

Well, $(A - 2 I )^3 = 0,$ but $(A - 2 I )^2 \neq 0.$ So the minimal polynomial and the characteristic polynomial agree, meaning each eigenvalue occurs in just a single block. Indeed $$ (A - 2 I )^2 = \left( \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 0 & 0 \\ -1 & -1 & -1 \end{array} \right) $$

Now find some $u$ such that $(A - 2 I )^2 u \neq 0.$ Since you want the extra $1$s below the main diagonal, we will put $u$ as the left hand column of $P,$ where we are solving $P^{-1} A P = J$ is the Jordan form you want. I will take $$ u = \left( \begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right) $$ The middle column will be $v = (A - 2 I) u,$ or $$ v = \left( \begin{array}{c} -2 \\ 1 \\ 1 \end{array} \right) $$ and finally $w = (A - 2 I) v =(A - 2 I)^2 u ,$ so that $ (A - 2 I) w =(A - 2 I)^3 u = 0 u = 0,$ so that $w$ is a genuine eigenvector $$ w = \left( \begin{array}{c} 1 \\ 0 \\ -1 \end{array} \right) $$ The columns of $P$ will be $u,v,w$ so $$ P = \left( \begin{array}{ccc} 1 & -2 & 1 \\ 0 & 1 & 0 \\ 0 & 1 & -1 \end{array} \right) $$ next $$ P^{-1} = \left( \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 1 & -1 \end{array} \right) $$

With $$ A = \left( \begin{array}{ccc} 0 & -1 & -2 \\ 1 & 3 & 1 \\ 1 & 0 & 3 \end{array} \right) $$ we get to $$ \left( \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 1 & -1 \end{array} \right) \left( \begin{array}{ccc} 0 & -1 & -2 \\ 1 & 3 & 1 \\ 1 & 0 & 3 \end{array} \right) \left( \begin{array}{ccc} 1 & -2 & 1 \\ 0 & 1 & 0 \\ 0 & 1 & -1 \end{array} \right) = \left( \begin{array}{ccc} 2 & 0 & 0 \\ 1 & 2 & 0 \\ 0 & 1 & 2 \end{array} \right) $$ $$ \left( \begin{array}{ccc} 1 & -2 & 1 \\ 0 & 1 & 0 \\ 0 & 1 & -1 \end{array} \right) \left( \begin{array}{ccc} 2 & 0 & 0 \\ 1 & 2 & 0 \\ 0 & 1 & 2 \end{array} \right) \left( \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 1 & -1 \end{array} \right) = \left( \begin{array}{ccc} 0 & -1 & -2 \\ 1 & 3 & 1 \\ 1 & 0 & 3 \end{array} \right) $$

If you would like to get the $1$s above the main diagonal instead, start over with the columns of $P$ in order $w,v,u,$ then calculate the new $P^{-1}$ Indeed, the fact that a single Jordan block is similar to its transpose gives a cheap proof that any matrix is similar to its transpose.