Question

In: Advanced Math

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.

Expert Solution

Solution

$A\in M_6(\mathbb{R}) \implies deg(P\lambda)$

we have $\hspace{2mm}A^3-4A^2+3A=0$

$\iff A\bigg(A^2-4A+3I\bigg)=0$

$\iff\bigg(A-I\bigg)\bigg(A-3I\bigg)=0$

Let $f(\lambda)=\bigg(\lambda-I\bigg)\bigg(\lambda-3I\bigg)$ Then $f(A)=0$

$\implies P(\lambda)=\bigg(\lambda-1\bigg)^{n_1}\bigg(\lambda-3\bigg)^{n_2} \hspace{2mm}$ where $n_1+n_2=6 \hspace{2mm}(1)$

we have $tr(A)=8\iff n_1(1)+n_2(3)=8$ $\iff n_1+3n_2=8$

from.(1) $\begin{cases} n_1+n_2=6 & \quad\\ n_1+3n_2=8 & \quad \end{cases} \implies 2n_2=2\implies n_2=1,n_1=5$

Answer

Therefore. $P(\lambda)=\bigg(\lambda-1\bigg)^5\bigg(\lambda-3\bigg)$

SUN BUNRA answered 2 years ago

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 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.

Jordan Canonical Form

Let A be a square matrix defined by

A =\begin{pmatrix}2&-3&1\\ 1&-2&1\\ 1&-3&2\end{pmatrix}

(a) Find the characteristic polynomial of A. (b) Show that A is diagonalizable then diagonalize it. (c) Write $A^n$ \hspace{2mm} in term of n.

Jordan Canonical Form

Let A be a square matrix defined by

A =\begin{pmatrix}-1&3&-1\\ -3&5&-1\\ -3&3&1\end{pmatrix}

(a) Find the characteristic polynomial of A. (b) Show that A is diagonalizable then diagonalize it. (c) Write

A^n

in term of n.

Jordan Canonical Form

Let A be a square matrix defined by

A = \begin{pmatrix}3&2\\ 3&-2\end{pmatrix}

(a) Find the characteristic polynomial of A. (b) Show that A is diagonalizable then diagonalize it. (c) Write

A^n

in term of n.

Question

Jordan Canonical Form

Solutions

Expert Solution

Related Solutions

Jordan Canonical Form

Jordan Canonical Form

Jordan Canonical Form

Jordan Canonical Form

Jordan Canonical Form

Jordan Canonical Form

Jordan Canonical Form

Jordan Canonical Form

Jordan Canonical Form

Jordan Canonical Form