# Finding eigenvalues and eigenvectors of a $3\times 3$ matrix

My matrix is $$A=\begin{pmatrix}0 & 0 & -2\\ 1 & 2 & 0 \\ 0 & -2 & 0\end{pmatrix}$$

I have to find its eigenvalues and eigenvectors but characteristic polynomial is $3$rd degree and I can't calculate it. Please give a help for it. Thanks.

-
Do you mean that you can't factor the polynomial or that you can't find it? In either case, what have you tried? –  EuYu Nov 12 '12 at 6:47
i have found the characteristic polynomial as x^3-2x^2-4=0, solved it with calculator and find x1= 2.5943 and others are complex roots. How can i make calculations in this way? –  aicha Nov 12 '12 at 6:54
Well, the roots that you've found are your three eigenvalues. For each eigenvalue, row reduce $A-\lambda I$ as usual to find your eigenvectors. There's nothing different than the usual routine, except maybe that the numbers are not as nice as you're used to. –  EuYu Nov 12 '12 at 7:02
as i understood,there is no way without making these calculations.i thought maybe there could be a different way to find eigenvectors. ok. thank you. –  aicha Nov 12 '12 at 7:12

The eigenvalues of a matrix A are the solutions $\lambda$ to the equation of the form $det(A-\lambda I)$. Det refers to the determinant of the matrix formed by $(A - \lambda I)$ and I is the $n$ x $n$ identity matrix -- this is called the characteristic equation. To find your eigenvalues and eigenvectors just find the solution using $det(A-\lambda I)$. If you need me to elaborate any further just ask.

-
Thank you, i know how to find eigenvalues and eigenvectors. My question is for this matrix which has 2.5943 as its eigenvalue. how to find eigenvector for this different value. it is too complicated. is there any other way,easier? –  aicha Nov 12 '12 at 8:32
You can convert your decimal value you got from your eigenvalue (2.5943) as a fraction. Then it will be a lot easier to find the eigenvector. Also, there are three eigenvalues. You have a complex conjugate as an eigenvalue too. –  diimension Nov 12 '12 at 8:38
Here is a youtube clip I just googled that shows how to convert a decimal into a fraction. Always try, if possible, to get fractions as your values. It will help a lot. youtube.com/watch?v=v6r-4q2oIpY –  diimension Nov 12 '12 at 8:40
when i convert it, it is 25943/10000 , isn't it? it is again complicated. but what if i write 2.6 instead of 2.5943 for the firs eigenvalue? at that time i can write 13/5 as fractional. Does it be true? Do i get correct answer? –  aicha Nov 12 '12 at 9:01
You should always use the exact numbers. I tried with your fractions , and yes, it is a bit messy but it is doable. But, if you do not want to go through the mess you can always use Mathematica. Have you heard about mathematica? It can provide you the eigenvectors you need within seconds. –  diimension Nov 12 '12 at 9:32