You are seeing the orbits of the matrix exponential map acting on a set of 26 vectors in \( \mathbb{R}^3 \).

## Exponential What?

The (usual) exponential function is a map \(\operatorname{exp} : \R \to \R \) given by the following power series: $$\exp(x) = \sum_{n=0}^\infty \frac{x^n}{n!}.$$ This power series can be shown to converge everywhere and has connections to Euler's constant \( e \). We can generalize this function to a map \( \operatorname{exp} : M_n(\R) \to M_n(\R) \), that is we apply the power series on real-valued square matrices: $$ \exp(X) = \sum_{n=0}^\infty \frac{X^n}{n!}. $$ This map can be shown to converge with respect to any matrix norm, but it is most easily shown with the operator norm as it is an easy consequence of the fact that the usual exponential function on \( \R \) converges everywhere.

## Ok, but what am I looking at?

The matrix exponential map has a suprising property: the exponential of

Formally speaking, a one parameter subgroup is a map \[ \varphi : \R \to G \] such that \( \varphi \) is a smooth homomorphism, in the sense that is it infinitely differentiable (this concept is defined since \( G \) is a smooth manifold by virtue of the fact that it is a Lie group, but this detail is not so important) and the following holds for all \( s, t \) in \( \mathbb R \) \[ \varphi(s + t) = \varphi(s) \varphi(t). \]

These one-parameter subgroups can be viewed geometrically as tracing a curve through the manifold \( G \). In our case, we a tracing a curve through the space of invertible \( 3 \times 3 \) matrices \( GL_3(\mathbb R) \). The connection between this and the matrix exponential is that every one-parameter subgroup of a matrix Lie group (i.e. a subgroup of \( GL_n(\mathbb R) \) which is still a Lie group) has the form \[ \varphi(t) = \exp(tA) \] for some matrix \( A \in M_n(\mathbb{R}) \).