## Evolution Through a Matrix Group

Because Matrix Groups are manifolds, they can be parametrized by a subspace of $$\mathbb{R}^n$$. We will investigate a subgroup of $$GL_2$$, the square 2x2 invertible matrices.

In the left panel, we see a velocity field, representing the initial velocity of a path through this subgroup. The input parameters control parameters of this initial velocity. On the right, a circle is deformed according to the path through GL2 traced out according to the initial velocity.

Play around with the parameters and take note of how differing the parameters results in a different final image! Try keeping the tmax fixed at various times and playing with the other four parameters.

x0: , y0: , tmax:

a:

b:

c:

d: