Jordan Canonical Form

Let A be a square matrix defined by \( A=\begin{pmatrix}4&-2&1\\ 2&0&1\\ 2&-2&3\end{pmatrix}\hspace{2mm} \)Find the minimal polynomial of A. Then express \( A^4 \) and \( A^{-1} \) in terms of A and I.

Solutions

Expert Solution

Solution

we have \( A=\begin{pmatrix}4&-2&1\\ 2&0&1\\ 2&-2&3\end{pmatrix} \) The characteristics polynomial is :

\( P(\lambda)=-\bigg(\lambda^3-S_1\lambda^2+S_2\lambda-s_3\bigg) \)

\( \bullet S_1=tr(A)=4+0+3=7 \)

\( \bullet S_2=\begin{vmatrix}4&-2\\ 2&0\end{vmatrix}+\begin{vmatrix}0&1\\ -2&3\end{vmatrix}+\begin{vmatrix}4&1\\ 2&3\end{vmatrix}=4+2+10=16 \)

\( \bullet S_3=\begin{vmatrix}4&-2&1\\ \:2&0&1\\ \:2&-2&3\end{vmatrix}=12 \)

\( \implies P(\lambda)=-\lambda^3+7\lambda^2-16\lambda+12=-\bigg(\lambda-2\bigg)^2\bigg(\lambda-3\bigg) \)

since. \( P(\lambda)=0\implies -\bigg(\lambda-2\bigg)^2\bigg(\lambda-3\bigg)=0 \iff \lambda_1=2,\lambda_2=3 \)

For \( \lambda_1=2 \) then \( \bigg(A-2I\bigg)=0 \)

\( \implies \begin{pmatrix}2&-2&1\\ 2&-2&1\\ 2&-2&1\end{pmatrix}\sim \begin{pmatrix}2&-2&1\\ 0&0&0\\ 0&0&0\end{pmatrix}\hspace{2mm} \)

Let \( x_1=t,x_2=S\hspace{2mm},t,S\in \mathbb{R} \)

\( \implies 2x_1-2x_2+x_3=0\implies x_3=-2t+2S \)

so, \( \hspace{2mm}x\begin{pmatrix}t\\ S\\ -2t+2S\end{pmatrix}=t\begin{pmatrix}1\\ 0\\ -2\end{pmatrix}+S\begin{pmatrix}0\\ 1\\ 2\end{pmatrix} \)

\( \implies E_{\lambda_1}=span\left\{\begin{pmatrix}1\\ \:0\\ \:-2\end{pmatrix},\begin{pmatrix}0\\ \:1\\ \:2\end{pmatrix}\right\} \)

\( \implies gm(\lambda_1)=am(\lambda_1)=2 \) and \( Index(\lambda_1)=1 \)

The minimal polynomial is :

\( m(\lambda)=\bigg(\lambda-2\bigg)\bigg(\lambda-3\bigg) \)

\( \bullet \) Express \( A^4 \) and \( A^{-1} \) interm of of A and I.

we have \( m(\lambda)=\bigg(\lambda-2\bigg)\bigg(\lambda-3\bigg)=\lambda^2-5\lambda+6 \)

Let \( x^4=\bigg(x^2-5x+6\bigg)4x+ax+b \)

\( \implies A^4=\bigg(A^2-5A+6I\bigg)4A+aA+bI \)

\( \iff A^4=aA+bI \)

\( \iff \begin{cases} x=3\implies 3^4=3a+b & \quad \\ x=2\implies 2^4=2a+b & \quad \end{cases}\hspace{2mm}\implies a=65,b=-114 \)

Thus, \( A^4=65A-114I \)

\( \bullet \)Find \( A^{-1} \)

\( \implies A^2-5A+6I=0\iff -A^2+5A=6I \)

\( \iff -\frac{1}{6}A\bigg(A-5I\bigg)=I \)

Therefore. \( A^{-1}=-\frac{1}{6}A\bigg(A-5I\bigg) \)

Answer

Therefore.

\( m(\lambda)=\bigg(\lambda-2\bigg)\bigg(\lambda-3\bigg) \)

\( A^{-1}=-\frac{1}{6}A\bigg(A-5I\bigg) \)

SUN BUNRA answered 2 years ago

Next > < Previous

Related Solutions

Jordan Canonical Form

Let \( A\in M_n(\mathbb{R})\hspace{2mm} \) and \( \hspace{2mm}\lambda_1, \lambda_2,...,\lambda_n \hspace{2mm} \)(no need distinct) be eigenvalues of A. Show that a). \( \sum _{i=1}^n\lambda _i=tr\left(A\right) \) b). \( \:\prod _{i=1}^n\lambda _i=\left|A\right|\: \)

Jordan Canonical Form

Let \( A\in M_n(\mathbb{R})\hspace{2mm} \) and\( \hspace{2mm} m_A(\lambda)\hspace{2mm} \) be its minimal polynomial. Let f be a polynomial satisfies\( \hspace{2mm}f(A) = 0. \hspace{2mm} \)Show that\( \hspace{2mm} f(\lambda) \hspace{2mm} \)is divisible by\( \hspace{2mm} m_A(\lambda). \)

Jordan Canonical Form

Determine the value of a so that \( \lambda = 2 \) is an eigenvalue of \( A=\begin{pmatrix}1&-1&0\\ a&1&1\\ 0&1+a&3\end{pmatrix} \) Then show that A is diagonallizable and diagonalize it.

Jordan Canonical Form

Let \( A\in M_6(\mathbb{R}) \) be an invertible matrix satisfies \( A^3-4A^2 + 3A = 0 \) and \( tr(A) = 8. \) Find the characteristics polynomial of A.

Jordan Canonical Form

Let A be a square matrix defined by \( A = \begin{pmatrix}-3&-1&-3\\ 5&2&5\\ -1&-1&-1\end{pmatrix} \) (a) Find the characteristic polynomial of A. (b) Find the eigenvalues of A. Show that A is not diagonalizable over \( \mathbb{R} \) (c) Show that A is diagonalizable over\( \mathbb{C} \). Find the eigenspaces. Diagonalize A. (d) Express \( A^n \) in the form of \( a_nA^2+b_ nA+c_nI_n \) where \( (a_n), (b_n) \) and \( (c_n) \) are real sequences to be specified....

Jordan Canonical Form

Let A be a square matrix defined by \( A = \begin{pmatrix}-2&-1&-5\\ 2&2&3\\ 4&2&2\end{pmatrix} \) (a) Find the characteristic polynomial of A. (b) Find the eigenvalues and eigenspaces of A. (c) Show that A is not diagonalizable, but it is triangularizable, then triangularize A. (d) Find the three real sequences \( (a)_n, (b)_n ,(c)_n \) satisfying. \( \begin{cases} a_{n+1}=-2a_n-b_n-5c_n \hspace{2mm},a_0=1 & \quad \\ b_{n+1}=2a_n+2b_n+3c_n \hspace{2mm}, b_0=0 & \quad \\ c_{n+1}=4a_n+2b_n+6c_n \hspace{2mm},c_0=1 & \quad \end{cases} \)

Jordan Canonical Form

Let A be a square matrix defined by \( A =\begin{pmatrix}-8&-3&-6\\ 4&0&4\\ 4&2&2\end{pmatrix} \) (a) Find the characteristic polynomial of A. (b) Find the eigenvalues and eigenspaces of A. (c) Show that A is not diagonalizable, but it is triangularizable, then triangularize A. (d) Write \( A^n \) in terms of \( I, A,A^2 \) and n.