Complex eigenvalues - solve the system

In summary, the conversation discusses finding the general solution for a system of differential equations using eigenvalues and eigenvectors. The person mentions their attempt at solving the problem and their confusion about which eigenvalue to use. They also make a mistake in their calculations but eventually arrive at the correct answer. They ask if there can be multiple general solutions to a problem and if they should only use the eigenvalue where 'i' is positive.
  • #1
schmiggy
38
0

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

ehild
 
  • #3
ehild said:
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
 
  • #4
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

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Related to Complex eigenvalues - solve the system

1. What are complex eigenvalues?

Complex eigenvalues are a type of eigenvalue that contains both a real and imaginary component. They are used in linear algebra to solve systems of equations and find the eigenvectors of a matrix.

2. Why do we need to solve systems using complex eigenvalues?

Sometimes, a system of equations may have complex solutions, and complex eigenvalues allow us to accurately represent and solve these systems. In addition, complex eigenvalues can also reveal important information about the behavior and stability of a system.

3. How do you find complex eigenvalues?

To find complex eigenvalues, you first need to set up a characteristic equation using the coefficients of the system of equations. Then, you can use the quadratic formula to solve for the eigenvalues, which may result in complex numbers.

4. Can complex eigenvalues have real eigenvectors?

Yes, complex eigenvalues can have real eigenvectors. In fact, a complex eigenvalue will always have at least one real eigenvector, and it may also have a complex eigenvector depending on the specific system of equations.

5. Are there any applications of complex eigenvalues in real life?

Yes, complex eigenvalues are used in various fields such as physics, engineering, and economics to model and analyze complex systems. Examples include analyzing the stability of electrical circuits and predicting the behavior of financial markets.

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