Question

Find the characteristics and the minimal polynomial of the following matrices over \( \mathbb{R} \), then deduce the their corresponding Jordan Canonical Form J.

B=\( \begin{pmatrix}2&-1&-1&2\\ 0&1&-1&2\\ 2&-5&-1&6\\ 1&-3&-2&6\end{pmatrix} \)

Answer 1

Solution

\( P(\lambda)=-\lambda^4+S_1\lambda^3+S_2\lambda^2+S_3\lambda+S_4 \)

          \( = \lambda^4-8\lambda^3+24\lambda^2-32\lambda+16=\bigg(\lambda-2\bigg)^4 \)

The minimail polynomial  Since \( P(\lambda)=\bigg(\lambda-2\bigg)^4 \) \( \implies \bigg(B-\lambda I\bigg)^k=0 \)

\( \bigg(B-2I\bigg)=\begin{pmatrix}0&-1&-1&2\\ 0&-1&-1&2\\ 2&-5&1&6\\ 1&-3&-2&4\end{pmatrix} \)

\( \bigg(B-2I\bigg)^2=\begin{pmatrix}0&-1&-1&2\\ 0&-1&-1&2\\ 2&-5&1&6\\ 1&-3&-2&4\end{pmatrix}\begin{pmatrix}0&-1&-1&2\\ 0&-1&-1&2\\ 2&-5&1&6\\ 1&-3&-2&4\end{pmatrix}=0 \)

Thus,\( m(\lambda)=\bigg(B-2I\bigg)^2 \) is the minimail polynomial deduce the jordan canonical form.

since. \( m(\lambda)=\bigg(B-2I\bigg)^2\implies index(2)=2 \)

\( E_{\lambda}=\bigg(B-\lambda I\bigg)x=0 \iff \bigg(B-2I\bigg)=0 \)

\( \begin{pmatrix}0&-1&-1&2\\ 0&-1&-1&2\\ 2&-5&1&6\\ 1&-3&-2&4\end{pmatrix}\hspace{2mm} \sim \hspace{2mm}\begin{pmatrix}1&0&11&-2\\ 0&1&1&-2\\ 0&0&0&0\\ 1&0&0&0\end{pmatrix} \)

let \( x_4=t,x_3=u\implies x_2=2t-u,x_1=2t-u \)

\( E_{\lambda}=\left\{\begin{pmatrix}2t-u\\ 2t-u\\ u\\ t\end{pmatrix}\in \mathbb{R}^2|t,u\in \mathbb{R}\right\} =span\left\{\begin{pmatrix}2\\ 2\\ 0\\ 1\end{pmatrix},\begin{pmatrix}-1\\ -1\\ 1\\ 0\end{pmatrix}\right\}\hspace{2mm}, dim(2)=gm(2)=2 \)

\( \)

Then,the Jordan Canonical form is 

\( J=\begin{pmatrix} 2&1&0&0\\ 0&2 \:&0&0\\ 0&0&2 \:&1\\ 0&0&0&2 \:\end{pmatrix} \hspace{2mm}or\hspace{2mm} J=\begin{pmatrix}2&1&0&0\\ 0&2&1&0\\ 0&0&2\:&1\\ 0&0&0&2\end{pmatrix} \)

 

Answer

Thus,\( m(\lambda)=\bigg(B-2I\bigg)^2 \) is the minimail polynomial deduce the jordan canonical form.

Then,the Jordan Canonical form is

\( J=\begin{pmatrix} 2&1&0&0\\ 0&2 \:&0&0\\ 0&0&2 \:&1\\ 0&0&0&2 \:\end{pmatrix} \hspace{2mm}or\hspace{2mm} J=\begin{pmatrix}2&1&0&0\\ 0&2&1&0\\ 0&0&2\:&1\\ 0&0&0&2\end{pmatrix} \)