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Back to the particular case of an Euclidean space, the spectrum of a linear operator on this space is the set of eigenvalues of its matrix, and so is a finite set. As long as the origin is not in the spectrum (the matrix is invertible), one obviously satisfies the path condition from the previous paragraph, and such, the theory implies that ln ''T'' is well-defined. The non-uniqueness of the matrix logarithm then follows from the fact that one can choose more than one branch of the logarithm which is defined on the set of eigenvalues of a matrix.
== A Lie group perspective ==
In the theory of [[Lie group|Lie groups]], there is an [[exponential map]] from a [[Lie algebra]] ''g'' to the corresponding Lie group ''G''
: <math> \exp : g \rightarrow G. </math>
For matrix Lie groups, the elements of ''g'' and ''G'' are square matrices and the exponential map is given by the [[matrix exponential]]. The inverse map <math> \log=\exp^{-1} </math> is mutivalued
and coincides with the matrix logarithm discussed here and maps from the Lie group ''G'' into the Lie algebra ''g''.
Note that the exponential map is a local diffeomorphism between a neighborhood ''U'' of the zero matrix <math> \underline{0} \in g</math> and a neighborhood ''V'' of the identity matrix <math>\underline{1}\in G</math>.
Thus the (matrix) logarithm is well-defined as a map
:<math> \log: V\subset G\rightarrow U\subset g</math>.
==See also==
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