1$\text{The characteristic polynomial of a function } T \text{ is } C_T(x) = ?$
$\det(xI - T_A)$
2$\text{The characteristic polynomial of a matrix } B \text{ is } C_B(x) = ?$
det(xI -B)
3How do you find the eigenvalues of T?
$\text{Solve } C_T(x) = 0 \text{ for x}$
$x's = \lambda_1, \lambda_2, \dots$
4$\text{What is the condition for a matrix } A \in M_{n \times n}(\mathbb{F}) \text{ to be diagonalisable?}$
$C_A(x) \text{ has } n \text{ solutions}$
5What is the condition for T to be diagonalisable?
$C_T(x) \text{ has } n \text{ solutions}$
6$\text{What is the method to find } P \in M_{n \times n}(\mathbb{F}) \text{ such that } P^{-1} A P \text{ is diagonal (} A \in M_{n \times n}(\mathbb{F}) \text{) ?}$
$1. \quad \text{Find the eigenvalues of } A: \lambda_1, \lambda_2, \dots$
$2. \quad \text{Find the corresponding eigenvectors } v_1, v_2, v_3, \dots$
$3. \quad P = [v_1, v_2, \dots]$
7What is an eigenvector of T?
$\text{A vector such that } T(v) = \lambda v$
8What is an eigenvalue of T?
$\lambda$ where $T(v) = \lambda v$
9True or false, A is diagionalisable if similar to a diagonal matrix D
10$\text{True or false, } T \text{ is diagonalizable if } \beta = \{v_1, v_2, \dots\} \text{ where } v_1, v_2, \dots \text{ are eigenvectors}$
11$\text{True or false, if } T \text{ is diagonalisable, then } T_A \text{ is diagonalizable?}$
12What is the method to find the eigenvectors of T?
$1. \quad \text{Find the eigenvalues of } T: \lambda_1, \lambda_2, \dots$
$2. \quad \text{Solve } (\lambda_1 I - T_A)v = 0, (\lambda_2 I - T_A)v = 0$
$3. \quad \text{Eigenvectors of } T \text{ are } v_1, v_2, \dots$
13What is an eigenspace of T?
14What is the method to find the eigenspaces of T?
$1. Find\text{ } T_A \text{ }= \text{ }A$
$2. \quad \text{Find eigenvalues of } A: \, \lambda_1, \lambda_2, \dots$
$3. \quad E_{\lambda_1} = null(\lambda_1I - A)$
$\quad \quad E_{\lambda_2} = null(\lambda_2I - A)$
$\quad \quad \dots$
15How do you check if T is diagonalisable?
$1. \quad \text{Find } T_A = A$
$2. \quad \text{Use the method to check if } A \text{ is diagonalizable}$
16What is true about the eigenvalues for two similar matrices A and B?
They have the same eigenvalues
17What is true about the determinant for two similar matrices A and B?
Their determinants are equal
18Can eigenvalues be 0?
Yes
19$\text{Can eigenvectors be }\underline0?$
No