Logarithm of a matrix: Difference between revisions

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==Existence==

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 <math> \{''z'' ∈\in '''\mathbb{C'''} |\ −π\vert \ -\pi < \textit{Im} \ ''z'' < π\pi \} </math>. 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. The existence of real matrix logarithms of real 2&times;2 matrices is considered in a later section.