Eigenvector of 3x3 matrix with complex eigenvalues

In summary, the conversation discusses finding eigenvalues and eigenvectors for a 3x3 matrix, with a focus on complex eigenvalues. The matrix A has eigenvalues of 0, 1+i, and 1-i. While the eigenvector for the eigenvalue of 0 can be found, there is confusion on how to find nonzero eigenvectors for the complex eigenvalues. The speaker also mentions getting the reduced row echelon form of a matrix, which may be related to their approach to finding the eigenvectors. However, it is clarified that the complex eigenvalues of 1+i and 1-i are not actually eigenvalues, leading to the question of how the speaker concluded they were.
  • #1
rayne1
32
0
Matrix A:
0 -6 10
-2 12 -20
-1 6 -10

I got the eigenvalues of: 0, 1+i, and 1-i. I can find the eigenvector of the eigenvalue 0, but for the complex eigenvalues, I keep on getting the reduced row echelon form of:
1 0 0 | 0
0 1 0 | 0
0 0 1 | 0

So, how do I find the nonzero eigenvectors of the complex eigenvalues?
 
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  • #2
rayne said:
Matrix A:
0 -6 10
-2 12 -20
-1 6 -10

I got the eigenvalues of: 0, 1+i, and 1-i. I can find the eigenvector of the eigenvalue 0, but for the complex eigenvalues, I keep on getting the reduced row echelon form of:
1 0 0 | 0
0 1 0 | 0
0 0 1 | 0

So, how do I find the nonzero eigenvectors of the complex eigenvalues?

Hi rayne!

It means that 1+i and 1-i are not actually eigenvalues.
How did you conclude they were?
 

1. What is an eigenvector?

An eigenvector is a vector that, when multiplied by a given matrix, returns a scalar multiple of itself. In other words, the direction of the eigenvector remains unchanged when multiplied by the matrix.

2. How do you find the eigenvectors of a 3x3 matrix with complex eigenvalues?

To find the eigenvectors of a 3x3 matrix with complex eigenvalues, you can use the characteristic polynomial method or the diagonalization method. The characteristic polynomial method involves finding the roots of the characteristic polynomial and then solving for the corresponding eigenvectors. The diagonalization method involves finding the diagonal matrix of eigenvalues and then using it to find the eigenvectors.

3. What is the importance of eigenvectors in 3x3 matrices with complex eigenvalues?

Eigenvectors are important in 3x3 matrices with complex eigenvalues because they provide insight into the behavior of the matrix. They can help us understand how the matrix will affect the direction of a vector and can also be used to find solutions to systems of linear equations.

4. Can a 3x3 matrix with complex eigenvalues have real eigenvectors?

Yes, a 3x3 matrix with complex eigenvalues can have real eigenvectors. This occurs when the complex eigenvalues are conjugate pairs, meaning that they have the same real part and opposite imaginary parts. In this case, the eigenvectors will also be conjugate pairs, with the same real part and opposite imaginary parts.

5. How are eigenvectors and eigenvalues related in a 3x3 matrix with complex eigenvalues?

Eigenvectors and eigenvalues are related in a 3x3 matrix with complex eigenvalues because the eigenvalues determine the scale factor for the corresponding eigenvectors. In other words, the eigenvalues tell us how much the eigenvectors will be stretched or compressed when multiplied by the matrix.

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