What do i do now? Eigan vectors wee

[mr_coffee]

hello everyone, I'm trying to find all the eigenvalues and eigenvectors. Then construct D and P such that A = PDP^-1;
A =
2 0 1
-1 3 -1
0 10 1
well when i took the determinant of $$A-\lambda = 0$$ I got:
$$\lambda^2-4\lambda+13$$ and got Eigenvalues of 2 +/- 6i;
but now I'm going to find the first eigenvector so i let
$$\lambda= 2+6i$$ I'm stuck on how I'm suppose to let x = a so i can get an eigenvector.
here is the rest of the work:http://img407.imageshack.us/img407/1271/lastscan0qp.jpg
thanks.

 

[HallsofIvy]

hello everyone, I'm trying to find all the eigenvalues and eigenvectors. Then construct D and P such that A = PDP^-1;
A =
2 0 1
-1 3 -1
0 10 1
well when i took the determinant of $$A-\lambda = 0$$ I got:
$$\lambda^2-4\lambda+13$$ and got Eigenvalues of 2 +/- 6i;
but now I'm going to find the first eigenvector so i let
$$\lambda= 2+6i$$ I'm stuck on how I'm suppose to let x = a so i can get an eigenvector.
here is the rest of the work:http://img407.imageshack.us/img407/1271/lastscan0qp.jpg
thanks.

How did you get a quadratic equation out of a 3 by 3 matrix? I come out with completely differerent eigenvalues. I got, as the eigenvalue equation,
$$-\lambda^3+ 6\lambda^2- 21\lambda+ 16k= 0$$
which has $$\lambda= 1$$ as one solution. Factoring $$(\lambda- 1)$$ out leaves
$$-\lambda^2+ 5\lambda- 16= 0$$ to be solved. That has complex solutions but the imaginary part is irrational.
By the way- your TEX wasn't showing properly because you were using
"\tex" to end rather than "/tex". I fixed that.

 

[mr_coffee]

Ahh thanks alot! Our professor couldn't figure this out, well he could, but he said he didn't want too, so he isn't making us solve it but thanks or clearing that up! I later went back and did it, its quite ugly.