Finding the directions of eigenvectors symmetric eigenvalue problem

In summary, eigenvectors are associated with eigenspaces and each eigenvalue has a 1D eigenspace. The vectors v are calculated using a normalised process, limiting the choice to ±v1, ±v2. In complex vector spaces, a normalised eigenvector can take the form αv, where v is a normalised eigenvector and α is any complex number of unit modulus. Real numbers of unit modulus reduce to ±1.
  • #1
Andrew1235
5
1
Homework Statement
In the symmetric eigenvalue problem, K~v=w2v where K~=M−1/2KM−1/2, where K and M are the stiffness and mass matrices respectively.
Relevant Equations
K~v=w2v where K~=M−1/2KM−1/2
In the symmetric eigenvalue problem, Kv=w^2*v where K~=M−1/2KM−1/2, where K and M are the stiffness and mass matrices respectively. The vectors v are the eigenvectors of the matrix K~ which are calculated as in the example below. How do you find the directions of the eigenvectors? The negatives of the eigenvectors of a matrix are also eigenvectors of the matrix.
 

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  • #2
Andrew1235 said:
How do you find the directions of the eigenvectors? The negatives of the eigenvectors of a matrix are also eigenvectors of the matrix.
When we talk about eigenvectors, we are really taking about eigenspaces. Each eigenvalue has an eigenspace of one or more dimensions associated with it. No single vector is the eigenvector. In this case, you have a 1D eigenspace associated with each eigenvalue.

The author has chosen normalised ##v_1, v_2##, which limits the choice to ##\pm v_1, \pm v_2##.

In complex vector spaces, a normalised eigenvector is determined only up to a complex "phase factor" of unit modulus. E.g. a normalised eigenvector can take the form ##\alpha v##, where ##v## is a normalised eigenvector and ##\alpha## is any complex number of unit modulus. And, of course, real numbers of unit modulus reduces to ##\pm 1##.
 

Related to Finding the directions of eigenvectors symmetric eigenvalue problem

1. What is a symmetric eigenvalue problem?

A symmetric eigenvalue problem is a mathematical problem that involves finding the eigenvalues and eigenvectors of a square matrix that is symmetric, meaning that it is equal to its own transpose.

2. Why is finding the directions of eigenvectors important in a symmetric eigenvalue problem?

In a symmetric eigenvalue problem, the eigenvectors represent the directions in which the matrix operates in a simple way. This can provide useful information about the behavior and properties of the matrix, making it an important aspect of the problem.

3. How do you find the directions of eigenvectors in a symmetric eigenvalue problem?

To find the directions of eigenvectors in a symmetric eigenvalue problem, you can use a variety of methods such as the power method, inverse iteration, or Jacobi method. These methods involve repeatedly multiplying the matrix by a vector and normalizing the resulting vector until it converges to an eigenvector.

4. What is the relationship between eigenvectors and eigenvalues in a symmetric eigenvalue problem?

In a symmetric eigenvalue problem, the eigenvectors and eigenvalues are closely related. Each eigenvector corresponds to a specific eigenvalue, and the eigenvalues represent the scaling factor by which the matrix operates on the eigenvector. The eigenvectors and eigenvalues are also orthogonal to each other.

5. Can a symmetric eigenvalue problem have multiple solutions?

Yes, a symmetric eigenvalue problem can have multiple solutions. In fact, there can be as many eigenvectors as there are dimensions in the matrix, and each eigenvector can have a different eigenvalue. This means that there can be an infinite number of solutions to a symmetric eigenvalue problem.

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