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gtse
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Homework Statement
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)
Homework 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.
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.