for some matrix , some vector and some scalar .
Then is called an eigenvector and is the corresponding Eigenvalue.
The thing is, how do you find and ?
It might make sense to move everything to one side.
Unfortunately is a matrix and is a scalar. So you instead write
Now that the matrices have the same dimensions, you can subtract them
Clearly, this has a trivial solution if .
For a nontrivial solution
Since this is a zero matrix, it is singular.
We can use this information to find .
In your case:
You need to work out
Since this determinant is zero...
You should be able to solve for now.