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!

 

[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!

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!