# Complex eigenvalues - solve the system

## Homework Statement

Using eigenvalues and eigenvectors, find the general solution to
dx/dt = x - y
dy/dt = x + y

## Homework 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.

## 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...

#### Attachments

• complex eigenvalues.jpg
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ehild
Homework Helper
Note that 1-(i+1)=-i instead of 2-i.

ehild

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

ehild

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

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??

#### Attachments

• fixed.jpg
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