How to Solve Complex Eigenvectors in Matrix Algebra

[tophman]

Hey,

I have a quick question that I can not seem to find much of an answer to in my text. When working with a nxn matrix, A, and you find eigenvalues that are complex, I'm confused about how to go about finding the actual eigenvector. I know we compute the null space of A-lambdaI, but that is where I seem to get stuck. For a 2x2, easy enough and I can do it. The problem is when n > 2. Gaussian elimination becomes a ridiculous mess. Is that the only way to do it? When I do substitution I end up with 0 = 0 which makes me think that each row is just some multiple of the other. If this is the case, do I just use any row I want?

Basically, I'm completely stuck with how to solve the complex matrix.

Any help would be greatly appreciated! :uhh:

 

[HallsofIvy]

Hey,

I have a quick question that I can not seem to find much of an answer to in my text. When working with a nxn matrix, A, and you find eigenvalues that are complex, I'm confused about how to go about finding the actual eigenvector. I know we compute the null space of A-lambdaI, but that is where I seem to get stuck. For a 2x2, easy enough and I can do it. The problem is when n > 2. Gaussian elimination becomes a ridiculous mess. Is that the only way to do it? When I do substitution I end up with 0 = 0 which makes me think that each row is just some multiple of the other. If this is the case, do I just use any row I want?

Basically, I'm completely stuck with how to solve the complex matrix.

Any help would be greatly appreciated! :uhh:

Well, of course, you get "0= 0". In order to be an eigenvalue, the equations you get with $\lambda$ equal to that eigenvalue, must be dependent so that 0 is not the only solution. I don't know what problem you are doing but Gaussian elimination is the best way to go- expect, of course, to use a TI-93 calculator that will do eigenvectors for you!

 

[tacman]

Hi there,

Solving for complex eigenvectors in matrix algebra can be a bit tricky, but there are a few methods that can help make the process easier. One method is to use the characteristic polynomial of the matrix, which is formed by taking the determinant of (A - λI). This polynomial can be solved for the eigenvalues, and then the corresponding eigenvectors can be found by substituting these values into (A - λI)x = 0 and solving for x.

Another approach is to use the Jordan canonical form, which involves finding a matrix P such that P^-1AP is in Jordan form. This form is useful because it has a simple structure that makes it easier to find eigenvectors. Once you have the Jordan form, you can use the same process as above to find the eigenvectors.

If you are using Gaussian elimination to solve for the eigenvectors, it can indeed become quite messy, especially for larger matrices. In this case, it may be helpful to use a computer program or calculator to assist with the computations.

In terms of using any row you want, it is important to remember that when solving for eigenvectors, you are looking for a non-zero vector that satisfies the equation (A - λI)x = 0. So, if you end up with 0 = 0 when substituting into the equation, it means that the eigenvector is a multiple of the other row. In this case, you can choose any non-zero multiple of that row as your eigenvector.

I hope this helps and provides some guidance for solving complex eigenvectors in matrix algebra. Don't hesitate to ask for further clarification or assistance if needed. Best of luck!