In the previous page you saw that once we have an eigenvector, every vector on the same line through the origin is also an eigenvector with the same eigenvalue:

You also saw the different arrangements of eigen-axes available for a $2 \times 2$ matrix. In this page, you will see how we use those eigen-axes to model what happens to every vector in $\mathbb{R}^2$.

Eigen-axes are incredibly useful for modeling what matrices do to vectors. To see this clearly, we'll start with the very nice case where the eigen-axes are the $x$-axis and $y$-axis.

Scaling the coordinates by the eigenvalues gives a very natural way to model what the matrix is doing.

Example 1: $\lambda_x = 4$ and $\lambda_y = 2$

Example 2: $\lambda_x = 1$ and $\lambda_y = -1$

Example 3: Matrix powers, $\lambda_x = 1.1$ and $\lambda_y = -\frac{3}{4}$