Finding the directions of eigenvectors symmetric eigenvalue problem

[Andrew1235]

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.

 

[PeroK]

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 $\pm 1$.