Nowadays, I'm studying for exponential map of Lie group.

my question is, To make the form of exp$\begin{pmatrix}x_{11}&x_{12}&\cdots \\x_{21}&\ddots \\ \vdots\end{pmatrix}$,

I have to multiply same matrixes. so I wonder more basically, when the matrix $A$ which has all component can be arbitrary number, although we make $A^N$,whether $A^N$ has also all component which can be arbitrary number or not.

additionally, My major is physics, by the way, I almost don't know about mathematic symbols. If you know the process of proof, please show me the proof without mathematic symbols possible.

  • $\begingroup$ Its not clear to me what you are asking. What do you mean by "$A^N$ also has all component which can be arbitrary or not"? The components of $A^N$ clearly depend on what the components of $A$ are. $\endgroup$ – Spencer Apr 23 at 6:51
  • $\begingroup$ Also how is the actual content of your question related to the question title? $\endgroup$ – Spencer Apr 23 at 6:51
  • $\begingroup$ because the exponential map of $GL(n,\mathcal{C})$ is made by \lim_{N\to \infty}[exp$\begin{pmatrix} x_{11}/N&x_{12}/N&\cdots \\ x_{21}/N&\ddots \end{pmatrix}$]^N. so, I think most general exponential map of Lie group can make all vector for fixed vector which is multiplied to exponential map. $\endgroup$ – 정재훈 Apr 23 at 7:00
  • $\begingroup$ Thank you for clarifying your question. $\endgroup$ – Spencer Apr 23 at 7:05
  • $\begingroup$ me too!! thank you for your interest!! $\endgroup$ – 정재훈 Apr 23 at 7:09

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