## Eigencurves

OMOOS

SOOS

OSOTS

OOOS

SOZS

Linear algebra is one of my favourite areas of mathematics. Its a simplification but you could say that the things that mathematics does well are small numbers and straight lines. The rest is just clever ideas to covert other things into those. As the mathematics of straight lines and flat surfaces, the importance of linear algebra should be clear. Its techniques are also very fast when done on a computer, allowing live motion in video games. This lead to every computer having a GPU, essentially a special chip for linear algebra.

Within linear algebra a central object is the linear transformation (that can be encoded as matrices) and the subspaces it preserves given by the eigenvectors. This gives some powerful tools to break linear transformations into pieces that can be studied more easily. As well as eigenvectors, however, linear transformations that do not have negative real eigenvalues preserve other families of curves. Curves that are taken to themselves by the transformation. These animations show how these curves change as the matrix changes. Giving a glimpse into the detail of what linear transformations do. These are of particular interest in dynamical systems where these images so some of behaviours possible close to a fixed point.

#### OMOOS

This animation shows a pair of eigenvalues changing from complex (creating a rotation and scaling) into real values. Can you tell where it happen? It uses the matrix

$\begin{bmatrix} 1 & -1\\ 1 & s \end{bmatrix}$

with s running from 1.05 to 4.05.

#### SOOS

This animation shows a hyperbolic fixed point (attracting in one direction and repelling in another) changes into a attracting equilibrium point, using the matrix

$\begin{bmatrix} s & 1\\ 1 & s \end{bmatrix}$

with s running from 1.05 to 4.05.

#### OSOTS

This animation shows a transformation changing as both the eigenvalues and the direction of the eigenvectors, using the matrix

$\begin{bmatrix} 1 & s\\ 1 & 2s \end{bmatrix}$

with s running from 0.05 to 3.05.

#### OOOS

This animation shows a transformation changing as both the eigenvalues and the direction of the eigenvectors, using the matrix

$\begin{bmatrix} 1 & 1\\ 1 & s \end{bmatrix}$

with s running from 1.05 to 4.05.

#### SOZS

This animation shows how a shear changes with different eigenvalues, using the matrix

$\begin{bmatrix} s & 1\\ 0 & s \end{bmatrix}$

with s running from 0.05 to 3.05.