Linear transformations and eigenvectors

This worksheet explores the effect of a linear transformation $T$ (in $\mathbb{R}^2$), and its relationship with the eigenvectors, eigenvalues and determinant.

Drag the point $P$ around the unit circle, and see how its image $T(P)$ changes.

Can you identify the eigenvectors and eigenvalues?

The large blue point is a point $P$ on the unit circle. Its image $T(P)$ under the linear transformation $T$ is shown as the smaller point.

Drag $P$ around the unit circle and see how the image $T(P)$ changes. Where are the eigenvectors? What (approximately) are the eigenvalues?

Click 'Show eigenvectors' at top-right to check your answer.

Click 'Show basis vectors' to see the effect of the transformation $T$ on the standard basis vectors ${\textbf{e}}_1, {\textbf{e}}_2$ (also called ${\textbf{i}},{\textbf{j}}$).

You can enter a new linear transformation by entering values in the matrix $T$ at top-left. You can also drag the images of the basis vectors $T({\textbf{e}}_1)$ and $T({\textbf{e}}_2)$ to change $T$.

Some interesting transformations to try:

  • $T = \begin{bmatrix} 0 & 1\\ 1 & 0\end{bmatrix}$
  • $T = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix}$ – enter this as $\begin{bmatrix} 0.71 & -0.71\\ 0.71 & 0.71\end{bmatrix}$
  • $T = \begin{bmatrix} 2 & 0\\ 0 & \frac{1}{2}\end{bmatrix}$
  • $T = \begin{bmatrix} \sqrt{2} & \frac{\sqrt{2}}{4} \\ \sqrt{2} & -\frac{\sqrt{2}}{4} \end{bmatrix}$ – enter this as $\begin{bmatrix} 1.41 & 0.35 \\ 1.41 & -0.35\end{bmatrix}$

Questions to consider:

  • What do the eigenvalues represent, geometrically?
  • What does the determinant represent, geometrically?
  • What is the relationship between the determinant ($\det T$) and the eigenvalues?
  • What does it mean geometrically if the determinant is negative? positive? zero?
  • Look at where (if anywhere) the image of the unit circle intersects with the unit circle. What is the significance of these intersection points? Under what conditions on the eigenvalues do the curves intersect?
  • What is the relationship between the images $T({\textbf{e}}_1)$, $T({\textbf{e}}_2)$ and the matrix $T$?

Other resources:

Geogebratube page for this applet

Inspired in part by the MATLAB program eigshow from the ATLAST project.