Motivating example for eigenvalues and eigenvectors

This example is based on example 4.1.1 from the Damiano, Little - A Course in Linear Algebra textbook.

Consider the linear transformation defined by the matrix in the standard basis. In other words, where is the standard basis of .

Let us take a look at the images of the basis vectors and : Indeed,

We can see that both basis vectors get rotated and stretched. Let us now choose a different basis for : and look at the transformation image of :

In matrix terms,

Exercise

Try to replicate the example with other matrices . Can you find examples where a “scaling” basis like can’t be found?

In other words, is just scaling in the direction of with a factor of , and leaves the orthogonal direction unchanged.

Question

How to look for and determine directions in which the transformation is scaling, and determine the scaling factor?

Eigenvalues and eigenvectors directly address this question.