Definition 17.14.1 (Matrix exponential). For , the matrix exponential is This converges absolutely for any . The exponential map sends .
Proposition 17.14.1 (Properties of the matrix exponential).
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If , then . In general, .
, confirming that is the tangent space at the identity.
The Baker–Campbell–Hausdorff formula: , where the higher-order terms involve iterated brackets. This shows the group multiplication near the identity is determined by the Lie bracket.
Note. The exponential map is a local diffeomorphism near (by the inverse function theorem, since ). It need not be surjective globally: for , the matrix is not in the image of . However, for compact connected Lie groups (e.g., ,), is surjective.