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 a lot! 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.