Eigenvector with Complex Eigenvalues - What am I doing wrong?

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Homework Help Overview

The discussion revolves around finding eigenvectors associated with complex eigenvalues, specifically addressing potential errors in the calculation process. The original poster expresses uncertainty about their approach, which they believe mirrors the method used for real eigenvectors.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the original poster's method of calculating eigenvectors, noting a potential mix-up in the order of components. There is also a mention of the relationship between the rows of the matrix and how to derive the eigenvector from it.

Discussion Status

Some participants have provided feedback on the original poster's calculations, pointing out specific errors and suggesting alternative approaches. The conversation appears to be productive, with participants exploring different interpretations of the problem without reaching a definitive consensus.

Contextual Notes

There is a mention of the original poster's method being similar to that used for real eigenvectors, which raises questions about the assumptions made in applying that method to complex eigenvalues. Additionally, the clarity of the original work is noted as a potential barrier to understanding.

alsvt
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Homework Statement


eigenvector.png

Homework Equations


Conjugate of a complex number
Matrix reduction

The Attempt at a Solution



My attempt is bordered. Sorry about the quality.
snapshot.jpg


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

\bmatrix<br /> 1 + i &amp; 1 \\<br /> -2 &amp; -1 + i \\<br /> \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).
 
Last edited:
@vela Thanks. I wasn't really thinking it through at the end.
 

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