Logarithm of a matrix: Difference between revisions

Revision as of 16:20, 10 September 2022

Revision as of 16:21, 10 September 2022

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Note that the exponential map is a local diffeomorphism between a neighborhood ''U'' of the zero matrix <math> \underline{0} \in \mathfrak{g}</math> and a neighborhood ''V'' of the identity matrix <math>\underline{1}\in G</math>.<ref>{{harvnb|Hall|2015}} Theorem 3.42</ref>

Thus the (matrix) logarithm is well-defined as a map,

:<math> \log: VG\subsetsupset V G\rightarrow U\subset \mathfrak{g}.</math>

An important corollary of [[Jacobi's formula]] then is