# Thread: Linear Algebra Eigenvectors -- question on x1+x2+x3 = 0 and more

1. ## Linear Algebra Eigenvectors -- question on x1+x2+x3 = 0 and more

Hello.

Okay, I'm struggling with eigenvectors -- I have already searched the forum and found a thread here (http://www.mathhelpforum.com/math-he...n-problem.html) which is more or less about the subject I'm interested in. But I still need to know how the answer was found.

Let A = [1 1 1]
----- = [0 0 0]
----- = [0 0 0]

This is a RREF matrix, which I want to 'extract' an eigenvector from...

TD! posted that the eigenvectors are e.g. (1,0,-1) and (1,-1,0). What I want to know is how they are found?

Also - I would like to know more about eigenvectors of this matrix...

B = [0 1 0]
- = [0 0 1]
- = [0 0 0]

I see x2 = x3 = 0, so I guess the eigenvector is [? 0 0], but what is x1? I think x1 is supposed to be x1 = 1, but why? I don't get why..?

Simon DK.

2. Originally Posted by sh01by
Hello.

Okay, I'm struggling with eigenvectors -- I have already searched the forum and found a thread here (http://www.mathhelpforum.com/math-he...n-problem.html) which is more or less about the subject I'm interested in. But I still need to know how the answer was found.

Let A = [1 1 1]
----- = [0 0 0]
----- = [0 0 0]

This is a RREF matrix, which I want to 'extract' an eigenvector from...

TD! posted that the eigenvectors are e.g. (1,0,-1) and (1,-1,0). What I want to know is how they are found?
it can be found by solving this equation:
det (xI - A) = 0, then solve for x..

Originally Posted by sh01by
Also - I would like to know more about eigenvectors of this matrix...

B = [0 1 0]
- = [0 0 1]
- = [0 0 0]

I see x2 = x3 = 0, so I guess the eigenvector is [? 0 0], but what is x1? I think x1 is supposed to be x1 = 1, but why? I don't get why..?

Simon DK.
actually, it is one of the eigenvectors. the eigenvectors of the matrix is in the set {[r 0 0] | r is in the Field}, this set is called the eigenspace of the eigenvalue of the matrix.

3. ## Hmm...

"it can be found by solving this equation:
det (xI - A) = 0, then solve for x.."

Isn't that the characteristic polynomial to find the eigenvalues?
If it is, then that is not really what I'm searching for... If it isn't, then I just don't understand entirely, maybe you could do an example?

4. Originally Posted by sh01by
"it can be found by solving this equation:
det (xI - A) = 0, then solve for x.."

Isn't that the characteristic polynomial to find the eigenvalues?
If it is, then that is not really what I'm searching for... If it isn't, then I just don't understand entirely, maybe you could do an example?
okay, after finding the eigenvalues, the eigenvectors are found by looking for the eigenspace, that is, solving for $\displaystyle det (\lambda I - A)X = 0$