Finding the eigenvalue given matrix A and an eigenvector?

• Apr 23rd 2013, 06:18 PM
lamentofking
Finding the eigenvalue given matrix A and an eigenvector?
The problem is attached. I got eigenvalues lambda1 = -1, lambda2 = -3, and lambda3 = -5. My textbook says the answer to this problem is lambda3 = -5. But when I try to plug in that eigenvalue myself I don't get the corresponding eigenvector given in the problem! I hope someone can assist me.
• Apr 23rd 2013, 06:36 PM
Gusbob
Re: Finding the eigenvalue given matrix A and an eigenvector?
I'm not sure what you mean...did you mean to say that when you're finding an eigenvector by solving $\displaystyle (A+5I)x=0$, you didn't get the vector v=[3,-2,1]? That's all right, because there are actually an infinite amount of eigenvectors, and they are precisely the set $\displaystyle \{cv|c\in \mathbb{R}\}$ where v=[3,-2,1]. So as long as you got a multiple of v, you're fine.

But judging from the question, all you had to do was left multiply your vector v by the matrix A and verify that Av=[-15,10,-5]=-5[3,-2,1]=-5v.
• Apr 23rd 2013, 08:15 PM
lamentofking
Re: Finding the eigenvalue given matrix A and an eigenvector?
Quote:

Originally Posted by Gusbob
But judging from the question, all you had to do was left multiply your vector v by the matrix A and verify that Av=[-15,10,-5]=-5[3,-2,1]=-5v.

By jove that is right! Thank you Gus!