One-parameter subgroup (${\gamma }_X(t)=\exp (tX)$)

A one-parameter subgroup of a Lie group $G$ is a smooth group homomorphism $\gamma :(\mathbb{R},+)\to G$: it satisfies $\gamma (s+t)=\gamma (s)\gamma (t)$, with $\gamma (0)=e$ and $\gamma (-t)=\gamma (t{)}^{-1}$. Every such map has the form $\gamma (t)=\exp (tX)$ for a unique $X={\gamma }^{\prime }(0)$ in the Lie algebra of $G$. Thus its initial tangent vector determines the whole motion, for all real times. The map need not be injective; the word subgroup describes its image. See MIT 18.745, Lecture 7, Proposition 7.1.

Differentiate the planar rotation matrix at angle zero:The even and odd powers in the matrix exponential therefore giveThis exact formula gives the whole one-parameter subgroup. Taking $X=J$ means unit angular velocity: at $t=\pi /2$, $\exp (tJ)=J$ sends $(1,0{)}^T$ to $(0,1{)}^T$, a counterclockwise quarter-turn. At $t=\pi$, it equals $-I$; at $t=2\pi$, it returns to $I$. Choosing $X=2J$ reaches the quarter-turn at $t=\pi /4$. Changing the generator changes the speed, while preserving the circular path.

For a moving vector $v(t)=R_{t}v(0)$, the ordinary differential equation $v^{\prime }(t)=Jv(t)$ says its velocity is perpendicular to its position. The beautiful compression is that one constant matrix specifies every angle of this endless family of rotations.

For $X\ne 0$, the image of $t\mapsto e^{tX}$, with its intrinsic immersed Lie subgroup topology, is isomorphic to $(\mathbb{R},+)$ or the circle group. The kernel is respectively zero or $\tau \mathbb{Z}$ for a least positive period $\tau$. The case $X=0$ is the trivial group. An irrational winding in a two-dimensional torus need not be an embedded or closed subgroup, so its intrinsic topology must be distinguished from the subspace topology.

For $X=\left(\begin{array}{cc}a& b\\ c& -a\end{array}\right)$, the image of its matrix exponential is compact precisely when $a^2+bc<0$ or $X=0$. If $a^2+bc=-{\omega }^2<0$, then $e^{tX}=\cos (\omega t)I_2+\sin (\omega t)X/\omega$ and its image is conjugate to the circle group. A nonzero nilpotent generator or a generator with real nonzero eigenvalues produces an unbounded image.