Questions
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.

In: Advanced Math

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. 

In: Advanced Math

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.

In: Advanced Math

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). \( \:\prod _{i=1}^n\lambda _i=\left|A\right|\: \)

 

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.

 

In: Advanced Math

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\( \hspace{2mm} m_A(\lambda). \)

In: Advanced Math

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 

\( m(\lambda)=\bigg(\lambda-1\bigg)^3\bigg(\lambda-3\bigg)^2 \)

In: Advanced Math

Determine all possible Jordan canonical forms for a linear operator L whose characteristic polynomial is

Determine all possible Jordan canonical forms for a linear operator L whose characteristic polynomial is 

\( P(\lambda)=\bigg(\lambda-3\bigg)^3\bigg(\lambda-4\bigg)^2 \)

In: Advanced Math

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=\( \begin{pmatrix}2&1&0&0\\ 0&2&0&0\\ 0&0&3&0\\ 0&0&0&3\end{pmatrix} \)

In: Advanced Math

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=\( \begin{pmatrix}2&0&0&0\\ 0&2&0&0\\ 0&0&1&0\\ 0&0&0&3\end{pmatrix} \)

In: Advanced Math

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

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.

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

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

In: Advanced Math

Find all eigenvalues and their corresponding eigenspaces of the following matrix.

Find all eigenvalues and their corresponding eigenspaces of the following matrix. B=\( \begin{pmatrix}2&-3&1\\ 1&-2&1\\ 1&-3&2\end{pmatrix} \)

In: Advanced Math

Find all eigenvalues and their corresponding eigenspaces of the following matrix.

Find all eigenvalues and their corresponding eigenspaces of the following matrix. A.\( \begin{pmatrix}3&0\\ 1&2\end{pmatrix} \)

In: Advanced Math