Is there more than one possibility for eigenvectors of a single eigenvalue

In summary, the conversation discusses the eigenvalues and eigenvectors of a given matrix. The matrix has two eigenvalues, i and -i, and the conversation explores a possible error in finding the corresponding eigenvector. It is discovered that the two eigenvectors calculated are scalar multiples of each other, which is a common occurrence in determining eigenvectors.
  • #1
mkerikss
18
0
I guess this is best explained with an example. The matrix (0 -1) has the eigenvalues
------------------------------------------------------------------ (1 0)
i and -i. For -i we obtain ix1-x2=0 and x1+ix2=0. I got a corresponding eigen vector (1 i), but when I controlled this result with wolfram alpha it gave me the vector (-i 1). By inserting these values in the equations both of these give 0, as they should, so my guess is they can both be used as the eigenvector for the eigenvalue -i, but I'm not sure though, as I haven't come across this kind of situation before. If somebody could help it would be most appreciated.
 
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  • #2
Note that those two vectors are scalar multiples of each other, and eigenvectors are determined up to scalar multiples.
 
  • #3
Ok, I noticed that now, if i multiply the first one with -i I get the second one and vice versa. Thank you :smile:
 

What does it mean for an eigenvalue to have multiple eigenvectors?

When an eigenvalue has multiple eigenvectors, it means that there are multiple vectors that can be multiplied by the corresponding matrix to produce the same scalar multiple of the original vector. In other words, these vectors are all transformed in the same way by the matrix, but they may have different magnitudes or directions.

Can a single eigenvalue have an infinite number of eigenvectors?

Yes, it is possible for a single eigenvalue to have an infinite number of eigenvectors. This occurs when the matrix is a scalar multiple of the identity matrix, where every vector is an eigenvector with the same eigenvalue.

Why is it important to find eigenvectors for a given eigenvalue?

Finding eigenvectors for a given eigenvalue is important because they provide a basis for the vector space and can be used to simplify calculations and solve systems of linear equations. Additionally, eigenvectors and eigenvalues play a crucial role in understanding the behavior of linear transformations and diagonalizing matrices.

Can two different eigenvalues have the same eigenvector?

No, two different eigenvalues cannot have the same eigenvector. Each eigenvector corresponds to only one eigenvalue, and vice versa. It is possible, however, for two different eigenvalues to have eigenvectors that are scalar multiples of each other.

How do you determine the number of eigenvectors for a given eigenvalue?

The number of eigenvectors for a given eigenvalue is equal to the geometric multiplicity of that eigenvalue, which is the dimension of the eigenspace associated with that eigenvalue. This can be determined by finding the null space of the matrix subtracted by the eigenvalue times the identity matrix.

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