##### 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 $$A=\begin{pmatrix}4&-2&1\\ 2&0&1\\ 2&-2&3\end{pmatrix}\hspace{2mm}$$Find the minimal polynomial of A. Then express $$A^4$$ and $$A^{-1}$$ in terms of A and I.

##### 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 $$A\in M_6(\mathbb{R})$$ 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 $$A\in M_n(\mathbb{R})\hspace{2mm}$$ 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$$\hspace{2mm} m_A(\lambda).$$

##### 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

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

##### 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

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

##### 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 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=$$\begin{pmatrix}1&1&0&0\\ -1&-1&1&0\\ 0&1&1&0\\ -1&-1&1&1\end{pmatrix}$$

##### 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 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=$$\begin{pmatrix}1&-1&-1\\ 0&0&-1\\ 0&1&2\end{pmatrix}\:$$

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

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

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