I'm trying to understand this proof of the claim that if A is a one-parameter subgroup of [itex]M_n(\mathbb C)[/itex], there exists a unique matrix X such that [itex]A(t)=e^{tX}[/itex]. (Theorem 2.13, page 37. Proof on page 38).(adsbygoogle = window.adsbygoogle || []).push({});

To prove uniqueness, just note that A'(0)=X.

To prove existence, the author defines

[tex]X=\frac{1}{t_0} \log A(t_0)[/tex]

and then shows that [itex]A(t)=e^{tX}[/itex] for all t in a dense subset of the real numbers. Continuity then implies that this identity holds for all real t.

My problem with this is that he doesn't clearly state what t_{0}is, so did he really define what X is? I mean, if this t_{0}does the job, it looks like we could have used t_{0}/2 instead, or any other positive real number that's smaller than t_{0}. I'm probably missing something obvious, and I'm hoping someone can tell me what that is.

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# One-parameter subgroups and exponentials

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