# Eigenvectors - principle lines of force

In the article about PCA and coavariance matrix I've read the following:

Finding the eigenvectors and eigenvalues of the covariance matrix is the equivalent of fitting those straight, principal-component lines to the variance of the data. Why? Because eigenvectors trace the principal lines of force, and the axes of greatest variance and covariance illustrate where the data is most susceptible to change.

I think that phrase "principal lines of force" is associated with untuitive understanding of linear transformation and eigenvectors. Can someone please provide the intuition and, perhaps, examples to get the grasp of this line?

The goal is to understand the line in the light of data transformation: $$X = RSD$$, where $$X$$ is the data matrix, $$R$$ is rotation matrix, $$S$$ is scaling matrix, and $$D$$ is white data matrix (unit variance, zero mean).

Hopefully, I understand PCA concepts and this question is more concerned with linear algebra in general.

Cited Article

• This "lines of force" analogy is uncommon, but your reference does a good job explaining what it means. – whuber Jun 9 at 19:14