Logarithm of a matrix: Difference between revisions

Revision as of 22:14, 9 March 2022

Revision as of 16:20, 10 September 2022

Line 200:

== A Lie group theory perspective ==

In the theory of [[Lie group]]s, there is an [[exponential map (Lie theory)|exponential map]] from a [[Lie algebra]] ''<math>\mathfrak{g''}</math> to the corresponding Lie group ''G''

: <math> \exp : \mathfrak{g} \rightarrow G. </math>

For matrix Lie groups, the elements of ''<math>\mathfrak{g''}</math> and ''G'' are square matrices and the exponential map is given by the [[matrix exponential]]. The inverse map <math> \log=\exp^{-1} </math> is multivalued and coincides with the matrix logarithm discussed here. The logarithm maps from the Lie group ''G'' into the Lie algebra ''<math>\mathfrak{g''}</math>.

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

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