Question

In: Advanced Math

Find the characteristics and the minimal polynomial of the following matrices over R, then deduce the their corresponding Jordan Canonical Form J.

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

A=\( \begin{pmatrix}1&-1&-1\\ 0&0&-1\\ 0&1&2\end{pmatrix}\: \)

Solutions

Expert Solution

Solution

A=\( \begin{pmatrix}1&-1&-1\\ 0&0&-1\\ 0&1&2\end{pmatrix}\: \) The characteristics polynomial.

\( \implies P(\lambda)=det(A-\lambda I)=-\bigg(\lambda^3 -S_1\lambda^2+S_2\lambda -S_3\bigg) \)

\( \bullet S_1=tr(A)=1+0+2=3 \)

\( \bullet S_2=\begin{vmatrix}1&-1\\ 1&2\end{vmatrix}+\begin{vmatrix}1&-1\\ 0&2\end{vmatrix}+\begin{vmatrix}1&-1\\ 0&0\end{vmatrix}=3 \)

\( \bullet S_3=\begin{vmatrix}1&-1&-1\\ 0&0&-1\\ 0&1&2\end{vmatrix}=1 \)

\( \implies P(\lambda )=-\bigg(\lambda^3-3\lambda^2+3\lambda-1\bigg)=-\bigg(\lambda-1\bigg)^3 \)

\( we\hspace{2mm} take P(\lambda)=0\iff -\bigg(\lambda-1\bigg)^3=0\implies \lambda =1\hspace{2mm}Thus,spact(A)=1 \)\( Let \hspace{2mm}x=\begin{pmatrix}x_1\\ x_2\\ x_3\end{pmatrix} \in E_1\hspace{2mm},Then\hspace{2mm}\bigg(A-I\bigg)x=0\hspace{2mm},for\hspace{2mm} x \neq 0 \)

\( \implies \begin{pmatrix}0&-1&-1\\ 0&-1&-1\\ 0&0&0\end{pmatrix} \sim \begin{pmatrix}0&1&1\\ 0&0&0\\ 0&0&0\end{pmatrix} \hspace{2mm}Let\hspace{2mm} x_3=S\hspace{2mm}and\hspace{2mm}x_1=t\in\mathbb{R} \)

\( \implies x_2+x_3=0\iff x_2=-x_3=-S \)

 

\( \implies x=\begin{pmatrix}t\\ -S\\ S\end{pmatrix}=t\begin{pmatrix}1\\ 0\\ 0\end{pmatrix}+S\begin{pmatrix}0\\ -1\\ 1\end{pmatrix}\hspace{2mm}Thus,\hspace{2mm}E_1=span\left\{\begin{pmatrix}1\\ 0\\ 0\end{pmatrix},\begin{pmatrix}0\\ -1\\ 1\end{pmatrix}\right\} \)

\( \iff m(\lambda)=\bigg(\lambda-1\bigg)^2\implies gm(1)=dim(E_1)=2 \)

Therefore. there are 2 jordan block and index(1)=2

\( \)


Answer :

Therefore. spact(A)=1 and  there are 2 jordan block and index(1)=2.

Related Solutions

Find the characteristics and the minimal polynomial of the following matrices over R , then deduce the their corresponding Jordan Canonical Form J.
Find the characteristics and the minimal polynomial of the following matrices over \( \mathbb{R} \), then deduce the their corresponding Jordan Canonical Form J. B=\( \begin{pmatrix}2&-1&-1&2\\ 0&1&-1&2\\ 2&-5&-1&6\\ 1&-3&-2&6\end{pmatrix} \)
Find the characteristics and the minimal polynomial of the following matrices over R, then deduce their corresponding Jordan Canonical Form J.
Find the characteristics and the minimal polynomial of the following matrices over R, then deduce their corresponding Jordan Canonical Form J. C=\( \begin{pmatrix}1&1&0&0\\ -1&-1&1&0\\ 0&1&1&0\\ -1&-1&1&1\end{pmatrix} \)  
Determine all possible Jordan canonical forms J for a matrix of order 6 whose minimal polynomial is
Determine all possible Jordan canonical forms J for a matrix of order 6 whose minimal polynomial is  \( m(\lambda)=\bigg(\lambda-1\bigg)^3\bigg(\lambda-3\bigg)^2 \)
The following are the Jordan Canonical Form of linear transformations. Find the characteristic polynomial, minimal polynomials, the algebraic multiplicity, geometric multiplicity and the index of each of the eigenvalues of L.
The following are the Jordan Canonical Form of linear transformations. Find the characteristic polynomial, minimal polynomials, the algebraic multiplicity, geometric multiplicity and the index of each of the eigenvalues of L. B=\( \begin{pmatrix}2&1&0&0\\ 0&2&0&0\\ 0&0&3&0\\ 0&0&0&3\end{pmatrix} \)
The following are the Jordan Canonical Form of linear transformations. Find the characteristic polynomial, minimal polynomials, the algebraic multiplicity, geometric multiplicity and the index of each of the eigenvalues of L.
The following are the Jordan Canonical Form of linear transformations. Find the characteristic polynomial, minimal polynomials, the algebraic multiplicity, geometric multiplicity and the index of each of the eigenvalues of L. A=\( \begin{pmatrix}2&0&0&0\\ 0&2&0&0\\ 0&0&1&0\\ 0&0&0&3\end{pmatrix} \)
Find the characteristics and characteristic coordinates, and reduce the following equation to canonical form: for y>0...
Find the characteristics and characteristic coordinates, and reduce the following equation to canonical form: for y>0 only Uxx+yUyy=0
For each of the following matrices, find a minimal spanning set for its Column space, Row...
For each of the following matrices, find a minimal spanning set for its Column space, Row space,and Nullspace. Use Octave Online to get matrix A into RREF. A = [4 6 10 7 2; 11 4 15 6 1; 3 −9 −6 5 10]
Find the reduced row echelon form of the following matrices. Interpret your result by giving the...
Find the reduced row echelon form of the following matrices. Interpret your result by giving the solutions of the systems whose augmented matrix is the one given. [ 0 0 3 -1 5 1 0 0 4 2 4 1 3 0 -8 1 2 7 9 0 ]
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT