# Complex eigenvalues - solve the system

1. Jan 26, 2013

### schmiggy

1. The problem statement, all variables and given/known data
Using eigenvalues and eigenvectors, find the general solution to
dx/dt = x - y
dy/dt = x + y

2. Relevant equations
Matrix 'A' - lambda*identity matrix ; for finding eigenvalues and thus eigenvectors

Other relevant equations written on the attached scanned image of my attempt at solving the question.

3. The attempt at a solution
Attached is my attempt, my lecture notes aren't clear on which eigenvalue to use when determining a general solution so at first I used the eigenvalue lambda = i + 1 which yielded a solution far from that in the answer section of this work booklet.

Using lambda = -i + 1 I got an answer very similar to the correct answer. At the bottom surrounded by a scribbled box is the answer from the book however I'm confused how they got their imaginary values...

#### Attached Files:

• ###### complex eigenvalues.jpg
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2. Jan 26, 2013

### ehild

Note that 1-(i+1)=-i instead of 2-i.

ehild

3. Jan 26, 2013

### schmiggy

Ugghhh how embarrassing.. sorry, stupid error. I'll fix that up in the morning.. how about the second lot of calculations though? I'm pretty sure I got the basic arithmetic right on them.. still the answer is wrong.
Thanks

4. Jan 26, 2013

### schmiggy

Ok, I've fixed my atrocious basic level maths from last night and got the correct answer.. I'm honestly not sure how I made such a terrible mistake but I did.. sorry if I wasted anyone's time.

My question remains however, is there only one general solution to a problem? For example using the eigenvalue i + 1 gives me the general solution as per the back of the book, however using the eigenvalue -i + 1 gives me an answer close to the one at the back but not quite.

When calculating a general solution are we supposed to only use the eigenvalue where 'i' is positive??

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