Eigenvector with Complex Eigenvalues - What am I doing wrong?

[alsvt]

Homework Statement

Homework Equations

Conjugate of a complex number
Matrix reduction

The Attempt at a Solution

My attempt is bordered. Sorry about the quality.

So I'm not sure what I'm missing. I use the exact same method that I use for normal eigenvectors, just with complex numbers in the mix.

 

[vela]

You messed up in the last step and swapped x₁ and x₂. If you let x₂=1, then your equation gave you x₁=-1/2+1/2 i. In the vector, however, you have the two values in the other order.

 

[hgfalling]

I can't quite read all of your work, but if you subtract $$\lambda$$ along the diagonal, you get:

$$\bmatrix 1 + i & 1 \\ -2 & -1 + i \\ \endbmatrix x = 0$$

The rows of this matrix are multiples (-1 + i) of each other, so you can use either row to find the eigenvector:

$$x_1 (1 + i) + x_2 = 0$$
$$-x_1 (1 + i) = x_2$$

so your vector is (1, -1 - i). This happens to not be one of your choices, but you can multiply it by i to obtain (i, 1 - i).

 

[alsvt]

@vela Thanks. I wasn't really thinking it through at the end.