A Demo for eigenvalues / eigenvectors:
shows that the eigenvectors of the covariance matrix for a set of point vectors represents the principal axes of the distribution and its eigen values are related with the lengths of the distribution along the principal axes. The difference among the eigenvalues determines how oblong the overall shape of the distribution is.
This is an example from book "Applied Numerical Methods Using MATLAB", page 387.
Thank you John for your very informative comment.
Hmm. An interesting idea here. I admit that long ago when I first took a linear algebra class as student, I learned the theory. But I did not learn what they mean, what the eigenvalues and eigenvectors tell you about a matrix. I did not learn those concepts until I saw eigenvalues and eigenvectors as they could be used in various applications, in various disciplines. I think this is true of many people. So this submission is a good one, IF it helps students to learn these concepts.
In fact, be careful, as this teaches you ONE aspect of this meaning, i.e., that of a covariance matrix. So you will learn something about what eigenvalues tell you about a covariance matrix. However, eigenvalues also appear in many other areas of mathematics, probability and statistics, engineering, science. For example, in mechanical engineering, you can learn about the principal stresses and strains of a system from the stress and strain matrices. In general, we will see that an eigenvector represents a "direction", and an eigenvalue represents the extent of "stretch" in that direction.
Anything that helps to teach these concepts is a good thing.