Calculating Eigenvalues and vectors for higher dimensions

Question

Sorry for these questions, I'm trying to understand and cannot seem to figure it out anywhere.

The material I've read so far on Eigenvalues and eigenvectors have showed for 2x2, 3x3.

From what I understand from the Eigenvalues, is that it shows how the data has been transformed (how much by) whereas the Eigenvectors show at what direction.

For a 2x2 matrix, I used the following:

$$\frac{a+b\pm\sqrt{(a+b)^2-4(ab-c^2)}}2.$$

This works fine, however, the problem that I'm facing is that I have a matrix of size: 451x128 which when I compute the Covariance matrix, I get the dimensions of 128x128.

My question is: How do I take the Eigenvalues/eigenvectors of a matrix (128x128) of this magnitude? I was lead to believe that, since, the Eigenvalues and eigenvectors show how to matrix has changed, there should be just a few values?

Hope someone can help

Answer 1

From the definition of an eigenvalue we have:

$$Ax=\lambda x$$ $$Ax-\lambda x =0$$ $$Ax-\lambda I x=0$$

where $I$ is the identity matrix. Then,

$$(A-\lambda I)x=0$$

Since we require that the eigenvector, $x$, be non-zero, the matrix $A-\lambda I$ must be singular. So we can compute the eigenvalues by solving the equation

$$det(A-\lambda I)=0$$

So with this being said, it would be very difficult for you to compute the eigenvalues of a very large matrix by hand. Matlab or Mathematica will have helpful routines for doing so.