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

  • Context: Undergrad 
  • Thread starter Thread starter mkerikss
  • Start date Start date
  • Tags Tags
    Eigenvalue Eigenvectors
Click For Summary
SUMMARY

The discussion centers on the concept of eigenvectors corresponding to a single eigenvalue, specifically for the matrix represented by (0 -1; 1 0), which has eigenvalues i and -i. The user identifies two eigenvectors, (1, i) and (-i, 1), for the eigenvalue -i, confirming that both vectors satisfy the eigenvalue equation. This illustrates that eigenvectors are not unique and can differ by scalar multiples, a fundamental property in linear algebra. The user concludes that scalar multiples of eigenvectors are valid representations for the same eigenvalue.

PREREQUISITES
  • Understanding of eigenvalues and eigenvectors in linear algebra
  • Familiarity with matrix representation and operations
  • Knowledge of complex numbers and their properties
  • Experience with computational tools like Wolfram Alpha for verification
NEXT STEPS
  • Study the properties of eigenvectors and eigenvalues in linear algebra
  • Learn about the geometric interpretation of eigenvectors
  • Explore the concept of linear combinations and scalar multiples in vector spaces
  • Investigate the application of eigenvalues in differential equations and stability analysis
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as data scientists and engineers working with eigenvalue problems in machine learning and systems analysis.

mkerikss
Messages
17
Reaction score
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.
 
Physics news on Phys.org
Note that those two vectors are scalar multiples of each other, and eigenvectors are determined up to scalar multiples.
 
Ok, I noticed that now, if i multiply the first one with -i I get the second one and vice versa. Thank you :smile:
 

Similar threads

Undergrad Can one find a matrix that's 'unique' to a collection of eigenvectors?
  • · Replies 33 ·
2
Replies
33
Views
2K
Undergrad The Orthogonality of the Eigenvectors of a 2x2 Hermitian Matrix
  • · Replies 13 ·
Replies
13
Views
4K
Undergrad Getting eigenvalues of an arbitrary matrix with programming
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Eigenvectors for degenerate eigenvalues
  • · Replies 7 ·
Replies
7
Views
10K
Undergrad Eigenvalues and Eigenvectors question
  • · Replies 16 ·
Replies
16
Views
2K
Undergrad Proof that if T is Hermitian, eigenvectors form an orthonormal basis
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad Question about an eigenvalue problem: range space
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Eigenvalues and eigenvectors of a non-symmetric matrix?
  • · Replies 2 ·
Replies
2
Views
3K
MHB Linear Maps and Eigenvalues/Eigenvectors
  • · Replies 1 ·
Replies
1
Views
2K
Why Are There Additional Eigenvalues and Eigenvectors of J3?
  • · Replies 17 ·
Replies
17
Views
2K