(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

This isn't really a question in particular.

I am doing my first Differential Equations course, and in the complex eigenvalues part, I am getting confused as to how to find the eigenvectors.

Example:

Solve for the general solution of:

x' = (1 -1)x (don't know how to type a matrix using latex sorry)

(5 -3)

2. Relevant equations

I know how to find the eigenvectors if there were real eigenvalues, since I've taken Linear Algebra and know that you can just simply reduce the matrix into Gauss-Jordan form.

3. The attempt at a solution

The eigenvalues are -1 +/- 2i (this is the easy part)

What confuses me is the next step:

The examples usually only plug in one eigenvalue (say -1+2i), which I don't know why.

And so the matrix will now look something like this;

(2-2i -1)

( 5 -2-2i)

What happens next I don't understand, usually the example would take

(2-2i)v1 - v2 = 0 if (v1,v2) was one of the eigenvectors

And it is around here I get the most confused.

- do I use v1 or v2 as the free variable?

- do I use the top or bottom row (5v1 + (-2-2i)v2 = 0) to find v1 and v2?

I have tried reading the books and the examples but they never show what they exactly do.

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# Homework Help: Finding the eigenvectors from complex eigenvalues

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