# Thread: Finding the eigenvector

1. ## Finding the eigenvector

Given a specific Eigenvalue $\displaystyle \lambda$, how do I find an eigenvector of a Matrix $\displaystyle A$?

Specifically, if $\displaystyle \lambda = 4$ and $\displaystyle A =$ $\displaystyle \left[\begin{array}{rrr}3&0&-1\\2&3&1\\-3&4&5\end{array}\right]$

By the way, is there a way to make my Matrix look a little neater/lined up?

2. If $\displaystyle \begin{pmatrix}x\\y\\z\end{pmatrix}$ is an eigenvector, then $\displaystyle \left(\begin{array}{rrr}3&0&-1\\2&3&1\\-3&4&5\end{array}\right)\begin{pmatrix}x\\y\\z\end{ pmatrix}\ =\ \lambda\begin{pmatrix}x\\y\\z\end{pmatrix}\ =\ 4\begin{pmatrix}x\\y\\z\end{pmatrix}.$

There will be infinitely many eigenvectors corresponding the the eigenvalue 4, so just pick one of them.

Originally Posted by Nerdfighter
By the way, is there a way to make my Matrix look a little neater/lined up?
Instead of centre-aligning them, right-align them (as I have done).

3. How Do I pick one? I obviously can't pick any numbers for x, y, and z.

4. I did not say pick any numbers for $\displaystyle x,y,z.$ Pick $\displaystyle x,y,z$ so that they satisfy the eigenvalue equation.

5. Originally Posted by TheAbstractionist
I did not say pick any numbers for $\displaystyle x,y,z.$ Pick $\displaystyle x,y,z$ so that they satisfy the eigenvalue equation.
What is a good method to find $\displaystyle x,y,z$ such that $\displaystyle x,y,z$ satisfies the eigenvalue equation?

Could I simply pick the zero vector to be an eigenvector?

6. Originally Posted by Nerdfighter
Given a specific Eigenvalue $\displaystyle \lambda$, how do I find an eigenvector of a Matrix $\displaystyle A$?

Specifically, if $\displaystyle \lambda = 4$ and $\displaystyle A =$ $\displaystyle \left[\begin{array}{rrr}3&0&-1\\2&3&1\\-3&4&5\end{array}\right]$

By the way, is there a way to make my Matrix look a little neater/lined up?
Solve the matrix equation

$\displaystyle A\vec v = \lambda \vec v \iff (A-\lambda I)\vec v =0$

Since your eigen value is 4 we get

$\displaystyle (A-4 I)\vec v =0$

$\displaystyle \begin{bmatrix} 3-4 && 0&& -1\\ 2 && 3-4 && 1 \\ -3 && 4 && 5-4 \\ \end{bmatrix}$

$\displaystyle \begin{bmatrix} -1 && 0&& -1\\ 2 && -1 && 1 \\ -3 && 4 && 1 \\ \end{bmatrix}$

putting this in reduced row form we get

$\displaystyle \begin{bmatrix} 1 && 0&& 1\\ 0 && 1 && 1 \\ 0 && 0 && 0 \\ \end{bmatrix}$

Solving the above homogenious equation gives

$\displaystyle z=t, y=-t,x=-t$ or

$\displaystyle \begin{bmatrix} -t \\ -t \\ t \end{bmatrix}=t \begin{bmatrix} -1 \\ -1 \\ 1 \end{bmatrix}$

So that is the eigen vector

p.s if you use the \begin{bmatrix} \end{bmatrix} tags it will auto format all of your columns

7. Thank you, that's what I needed!

8. Originally Posted by Nerdfighter
Could I simply pick the zero vector to be an eigenvector?

No, the zero vector is never an Eigenvector, as if it was every scalar would be an Eigenvalue. An Eigenvector for the corresponding Eigenvalue $\displaystyle \lambda$ is defined to be a "non-zero vector such that $\displaystyle Mv = v \lambda$".

9. Originally Posted by Swlabr
No, the zero vector is never an Eigenvector, as if it was every scalar would be an Eigenvalue. An Eigenvector for the corresponding Eigenvalue $\displaystyle \lambda$ is defined to be a "non-zero vector such that $\displaystyle Mv = v \lambda$".
Ooh, very good to know. Thank you.