Fredrik
Jan14-09, 02:56 AM
I'm trying to understand this (http://books.google.com/books?id=m1VQi8HmEwcC&pg=PP1&dq=brian+c+hall&hl=sv#PPA38,M1) proof of the claim that if A is a one-parameter subgroup of M_n(\mathbb C), there exists a unique matrix X such that A(t)=e^{tX}. (Theorem 2.13, page 37. Proof on page 38).
To prove uniqueness, just note that A'(0)=X.
To prove existence, the author defines
X=\frac{1}{t_0} \log A(t_0)
and then shows that A(t)=e^{tX} 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 t0 is, so did he really define what X is? I mean, if this t0 does the job, it looks like we could have used t0/2 instead, or any other positive real number that's smaller than t0. I'm probably missing something obvious, and I'm hoping someone can tell me what that is.
To prove uniqueness, just note that A'(0)=X.
To prove existence, the author defines
X=\frac{1}{t_0} \log A(t_0)
and then shows that A(t)=e^{tX} 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 t0 is, so did he really define what X is? I mean, if this t0 does the job, it looks like we could have used t0/2 instead, or any other positive real number that's smaller than t0. I'm probably missing something obvious, and I'm hoping someone can tell me what that is.