Yont's question at Yahoo Answers (eigenvector)

[Fernando Revilla]

Here is the question:

The matrix A = [{-7,3},{0,-4}] has eigenvalues -4 and -7. What are the associated eigenvectors with -7. I figured out that -4 is with (1,1) but i can't figure out -7

Here is a link to the question:

Find the eigenvector? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[jakncoke1]

$\begin{bmatrix} -7 & 3 \\ 0 & -4 \end{bmatrix} * x = \begin{bmatrix}7x \\ 7 y\end{bmatrix}$

or

-7x + 3y = -7x or 3y = 0, or y = 0so vectors of the form , $\begin{bmatrix} x \\ 0 \end{bmatrix} $
with $x \not = 0 $
(Sorry fernando if i hijacked this, i know how much you love answering these but I've had 6 cups of coffee and i need to do something from going crazy)

 

[Fernando Revilla]

$\begin{bmatrix} x \\ 0 \end{bmatrix} $

Certainly, by definition, $ \begin{bmatrix}{x}\\y\end{bmatrix}\neq \begin{bmatrix}{0}\\0\end{bmatrix}$ is an eigenvector of $A$ associated to $\lambda=-7$ if and only if:

$\begin{bmatrix}{-7}&{\;\;3}\\{\;\;0}&{-4}\end{bmatrix}\begin{bmatrix}{x}\\y\end{bmatrix}=(-7)\begin{bmatrix}{x}\\y\end{bmatrix}$

You'll easily get the system $\{y=0$ so, all the eigenvalues associated to $\lambda=-7$ are

$\begin{bmatrix}{\alpha}\\0\end{bmatrix}$ with $\alpha\neq 0$.

 

[jakncoke1]

Certainly, by definition, $ \begin{bmatrix}{x}\\y\end{bmatrix}\neq \begin{bmatrix}{0}\\0\end{bmatrix}$ is an eigenvector of $A$ associated to $\lambda=-7$ if and only if:
$\begin{bmatrix}{-7}&{\;\;3}\\{\;\;0}&{-4}\end{bmatrix}\begin{bmatrix}{x}\\y\end{bmatrix}=(-7)\begin{bmatrix}{x}\\y\end{bmatrix}$.

You'll easily get the system $\{y=0$ so, all the eigenvalues associated to $\lambda=-7$ are

$\begin{bmatrix}{\alpha}\\0\end{bmatrix}$ with $\alpha\neq 0$.

yes sir, i fixed it. you are absolutely right.

 

[CellCoree]

Hello Yont,

Thank you for your question about eigenvectors. As you correctly pointed out, the eigenvalue -4 is associated with the eigenvector (1,1). To find the eigenvector associated with -7, we can use the following formula:

Av = λv

Where A is the matrix, v is the eigenvector, and λ is the eigenvalue. In this case, we have:

[-7,3][x,y] = -7[x,y]

Simplifying this equation, we get:

-7x + 3y = -7x

3y = 0

This means that y can be any value, as long as x is 0. Therefore, the eigenvector associated with -7 is [0,1].

I hope this helps answer your question. Keep up the good work in exploring eigenvectors and their properties!

Best,
[Your name]