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Thatsme314 (talk | contribs) m →Existence: The real-valued 2x2 case was written by User:Rgdboer and last appeared in the [22:01, 28 January 2021](https://en.wikipedia.org/w/index.php?title=Logarithm_of_a_matrix&oldid=1003417720), before someone deleted it Tag: Reverted |
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The question of whether a matrix has a logarithm has the easiest answer when considered in the complex setting. A complex matrix has a logarithm [[if and only if]] it is [[invertible matrix|invertible]].<ref>{{harvtxt|Higham|2008}}, Theorem 1.27</ref> The logarithm is not unique, but if a matrix has no negative real [[eigenvalue]]s, then there is a unique logarithm that has eigenvalues all lying in the strip {''z'' ∈ '''C''' | −π < Im ''z'' < π}. This logarithm is known as the ''principal logarithm''.<ref>{{harvtxt|Higham|2008}}, Theorem 1.31</ref>
The answer is more involved in the real setting. A real matrix has a real logarithm if and only if it is invertible and each [[Jordan block]] belonging to a negative eigenvalue occurs an even number of times.<ref>{{harvtxt|Culver|1966}}</ref> If an invertible real matrix does not satisfy the condition with the Jordan blocks, then it has only non-real logarithms. This can already be seen in the scalar case: no branch of the logarithm can be real at -1
==Properties==
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