Jordan decomposition, finding $S$

Question

let $A = \left( \begin{array}{} 1 & 0 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{array} \right)$

$A$ has one eigenvalue $\lambda=1$ with algebraic multiplicity $3$ but it has only two linearly independent eigenvectors $\left( \begin{array}{} 1 \\ 0 \\ 0 \end{array} \right),\left( \begin{array}{} 0 \\ 1 \\ 0 \end{array} \right)$(this can be easily checked since $A-\lambda I=A-I=\left( \begin{array}{} 0 & 0 & 3 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \end{array} \right)$ and both vectors are in $ker(A-I)$), so the geometric multiplicity is $2$ which is less than 3, the algebraic multiplicity.

and from here, I was told that I need to compute something like $(A-I)w=v$ to find $w$ where $v$ is the eigenvector. but I have two $v$'s, so I don't know which $v$ I should take.

and as I put $A$ in the Wolfram Alpha, I found its Jordan decomposition and there was no $\left( \begin{array}{} 1 \\ 0 \\ 0 \end{array} \right)$ in the column of $S$, such that $A=SJS^{-1}$, where $J$ is the Jordan form.

exactly how to find $S$? I'm so confused.

Answer 1

Hint: the Jordan form of $A$ is given by $$ J= \begin{pmatrix} 1 & 0 & 0 \cr 0 & 1 & 1 \cr 0 & 0 & 1 \end{pmatrix}. $$ Then the matrix equation $AS=SJ$ can be easily solved by linear equations in the coefficients of $S$, e.g., $$ S= \begin{pmatrix} 3 & 3 & 0 \cr 3 & 2 & 0 \cr 0 & 0 & 1 \end{pmatrix}. $$

Answer 2

After you've found the eigenvectors, the next step is to find vectors $v$ for which $(A-I)v\ne0$ and $(A-I)^2v=0$ (this basically means that $(A-I)v$ is an eigenvector. For the given matrix $A$ we have $$ (A-I)^2=\begin{pmatrix} 0&0&3\\0&0&2\\0&0&0 \end{pmatrix}^2=\begin{pmatrix} 0&0&0\\0&0&0\\0&0&0 \end{pmatrix}, $$ so any vector $v$ such that $(A-I)v\ne0$ would work. Take $v=(0,0,1)^T$, it'll be our first basis vector. Then take $u=(A-I)v=(3,2,0)^T$, it'll be the second basis vector. And for the third one we can take any other eigenvector which isn't collinear with $u$. Take $w=(1,0,0)^T$. So, the matrix $$ S=\begin{pmatrix} 0&3&1\\0&2&0\\1&0&0 \end{pmatrix} $$ is such that $$ A=S\begin{pmatrix} 1&0&0\\1&1&0\\0&0&1 \end{pmatrix}S^{-1}. $$