# Section 7.1 $\quad$ Eigenvalues and Eigenvectors

Let $L: V\to V$ be a linear transformation. $L$ maps a vector $\mathbf{v}$ in $V$ to another vector $L(\mathbf{v})$ in $V$.<br /><br /><br /><br />

Let $L: V\to V$ be a linear transformation. The number $\lambda$ is called<br /><br /><br /><br />

Note that a  linear transformation can be expressed in terms of a matrix:<br /><br /><br /><br />

>**Definition** Let $A$ be an $n\times n$ matrix.<br /><br /><br /><br />

### Example 1

Find the eigenvalues and the associated eigenvectors of the matrix $A$, where
\begin{equation*}
  A = \left[
        \begin{array}{cc}
          1 & 1 \\
          -2 & 4 \\
        \end{array}
      \right]
\end{equation*}


```python
from sympy import *

A = Matrix([[1, 1], [-2, 4]]);

A.eigenvects()
```




    [(2, 1, [Matrix([
       [1],
       [1]])]), (3, 1, [Matrix([
       [1/2],
       [  1]])])]



>**Definition** Let $A = [a_{ij}]$ be an $n\times n$ matrix.<br /><br /><br /><br />

>**Theorem** Let $A $ be an $n\times n$ matrix.<br /><br /><br /><br />

### Example 2

Let
\begin{equation*}
  A =
  \left[
    \begin{array}{ccc}
      1 & 2 & -1 \\
      1 & 0 & 1 \\
      4 & -4 & 5 \\
    \end{array}
  \right]
\end{equation*}
Find all eigenvalues and associated eigenvectors of $A$.


```python
from sympy import *

A = Matrix([[1, 2, -1], [1, 0, 1], [4, -4, 5]]);

A.eigenvects()
```




    [(1, 1, [Matrix([
       [-1/2],
       [ 1/2],
       [   1]])]), (2, 1, [Matrix([
       [-1/2],
       [ 1/4],
       [   1]])]), (3, 1, [Matrix([
       [-1/4],
       [ 1/4],
       [   1]])])]



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ipynb
Jupyter Notebook
Jupyter_Notes/Lecture33_Sec7-1_EigenvaluesEigenvectors.ipynb
xiuquan0418/MAT341
2fb7ec4e5f0771f10719cb5e4a00a7ab07c49b59
MIT
Jupyter_Notes/Lecture33_Sec7-1_EigenvaluesEigenvectors.ipynb
xiuquan0418/MAT341
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MIT
Jupyter_Notes/Lecture33_Sec7-1_EigenvaluesEigenvectors.ipynb
xiuquan0418/MAT341
2fb7ec4e5f0771f10719cb5e4a00a7ab07c49b59
MIT
Qwen/Qwen-72B
1. YES
2. YES