chapter3.md

title	description
Eigenvalue Eigenvector Problems	undefined

What are eigenvalue eigenvectors?

type: TabExercise

xp: 

key: edebfb1a2c

---

type: NormalExercise

xp: 

key: 5cac61dd4f

@instructions

How do we compute Eigenvectors and values

1. The definition of an eigenvalue is $\mathbf{A}\mathbf{x} = \lambda \mathbf{x}$

2. Rearrange to give $(\mathbf{A}-\lambda\mathbf{I})\mathbf{x}=\mathbf{0}$

3. For any non-trivial solution $(\mathbf{A}-\lambda\mathbf{I})$ must be non-invertible (this indicates that the determinent is 0).

4. Thus, $\det{(\mathbf{A}-\lambda\mathbf{I})}=0$. The determinent generates the Characteristic Polynomial of the matrix.

Once you have solved this problem then you can compute the eigenvectors.

As the Characteristic Polynomial is a $N$ degree polynomial it has $N$ roots though these roots need not be necessarily distinct. So a $N\times N$ matrix has $N$ (though not necessarily distinct) eigenvalues (or eigenvectors).

EXAMPLE: Computing the eigenvalues of a 2x2 martix

Lets find eigenvectors and eigenvalues for the matrix:

$$ \left( \begin{array}{cc} 1 & 2 \\ 3 & 2 \end{array} \right) $$

To find the eigenvalues we need to solve the problem: $$ \left| \begin{array}{cc} 1-\lambda & 2 \ 3 & 2-\lambda \end{array} \right|=0 $$

or $(1-\lambda)(2-\lambda)-6=0$ which after some rearrangement is $(\lambda-4)(\lambda+1)=0$. Solutions for this are $\lambda=4 \mbox{ or } -1$.

@hint

@pre_exercise_code

import numpy as np

@sample_code

A = np.array([[1,2],[3,2]])
print( np.linalg.eigvals(A) )

@solution

A = np.array([[1,2],[3,2]])
print( np.linalg.eigvals(A) )

@sct

Ex().check_object('A').has_equal_value()
success_msg('Great job!')