Quick question about Eigenvectors

In summary, every eigenvalue will have at least one eigenvector, as defined by the property that it satisfies the equation Av= \lambda v. However, it is possible for a repeated eigenvalue to have less linearly independent eigenvectors than the number of repetitions of that eigenvalue, known as its geometric multiplicity. Despite this, there will always be at least one eigenvector for each eigenvalue, with an infinite number existing for each eigenvalue forming a subspace. This is known as the difference between algebraic and geometric multiplicity.
  • #1
sjeddie
18
0
Is at least one eigenvector guaranteed to exist given that we have found at least one eigenvalue? So, for example, given that we have found an eigenvalue of multiplicity 2 of a matrix, are we guaranteed to find at least 1 eigenvector of that matrix? Why or why not?
 
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  • #3
Thanks.

So it is possible to have less eigenvectors than eigenvalues (according to wiki's explanation on geometric multiplicity), so is it possible to have no eigenvectors at all for some eigenvalues?
 
  • #4
Not quite. Every eigenvalue will have at least one eigenvector. It is possible, however, for a repeated eigenvalue to have less linearly independent eigenvectors (its geometric multiplicity) than the number of repetitions of that eigenvalue (its algebraic multiplicity). There will always be at least one eigenvector though.
 
  • #5
thank you very much rochfor1!
 
  • #6
The definition of "eigenvalue" is "[itex]\lambda[/itex] is an eigenvalue for linear operator A if and only if there exist a non-zero vector, v, such that [itex]Av= \lambda v[/itex]".

Such a vector is, of course, an eigenvector so, by definition, there exist at least one eigenvector corresponding to any eigenvalue. And, in fact, any multiple of an eigenvector or any linear combination of eigevectors corresponding to the same eigenvalue is also an eigenvector- there necessarily exist an infinite number of eigenvectors corresponding to any eigenvalue- they form a subspace.

Rochfor1 is specifically talking about the number of independent eigenvectors corresponding to each eigenvalue- the dimension of that subspace. That's the "geometric multiplicity" of that eigenvalue.
 

1. What are Eigenvectors?

Eigenvectors are a type of vector that are associated with a linear transformation of a vector space. They represent the directions along which a transformation has a simple effect, such as stretching or compressing.

2. How are Eigenvectors calculated?

Eigenvectors are calculated by finding the roots of the characteristic equation of a given matrix. The resulting values are the eigenvalues, and the corresponding eigenvectors are found by solving a system of linear equations.

3. Why are Eigenvectors important?

Eigenvectors are important because they provide a way to simplify complex transformations and make calculations more efficient. They also have various applications in fields such as physics, engineering, and computer science.

4. Can a matrix have multiple Eigenvectors?

Yes, a matrix can have multiple eigenvectors. In fact, most matrices have multiple eigenvectors corresponding to different eigenvalues. This is because a matrix can have different transformations along different eigenvectors.

5. How are Eigenvectors used in data analysis?

Eigenvectors are commonly used in data analysis, specifically in techniques such as Principal Component Analysis (PCA). In PCA, eigenvectors are used to transform a dataset into a lower-dimensional space while preserving the most important information. This is useful for tasks such as data visualization and dimensionality reduction.

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