Logarithm of a matrix: Difference between revisions

Revision as of 17:56, 6 March 2024

Revision as of 23:38, 16 April 2024

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(\alpha+2\pi n)^4~I_2

</math><br>

…... <br>

<math>

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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 <math> \{z \in \mathbb{C} \ \vert \ -\pi < \textit{Im} \ z < \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;22×2 matrices is considered in a later section.

==Properties==

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==Further example: Logarithm of rotations in 3D space==

A rotation {{mvar|R}} ∈ SO(3) in ℝ³<math>\mathbb{R}</math>³ is given by a 3&times;33×3 [[orthogonal matrix]].

The logarithm of such a rotation matrix {{mvar|R}} can be readily computed from the antisymmetric part of [[Rodrigues' rotation formula]], explicitly in [[Axis–angle representation#Log map from SO.283.29 to so.283.29|Axis angle]]. It yields the logarithm of minimal [[Frobenius norm]], but fails when {{mvar|R}} has eigenvalues equal to &minus;1 where this is not unique.

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:Let

::<math> A' = V^{-1} A V.\, </math>

:Then ''A&{{prime;|A}}'' will be a diagonal matrix whose diagonal elements are eigenvalues of ''A''.

:Replace each diagonal element of ''A&{{prime;|A}}'' by its (natural) logarithm in order to obtain <math> \log A' </math>.

:Then

::<math> \log A = V ( \log A' ) V^{-1}. \, </math>

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:<math>\log (\det(A)) = \mathrm{tr}(\log A)~. </math>

==Constraints in the 2 &times;× 2 case==

If a 2 × 2 real matrix has a negative [[determinant]], it has no real logarithm. Note first that any 2 × 2 real matrix can be considered one of the three types of the complex number ''z'' = ''x'' + ''y'' ε, where ε²² ∈ { −1, 0, +1 }. This ''z'' is a point on a complex subplane of the [[ring (mathematics)|ring]] of matrices.<ref>{{Wikibooks-inline|Abstract Algebra/2x2 real matrices}}</ref>

The case where the determinant is negative only arises in a plane with ε²² =+1, that is a [[split-complex number]] plane. Only one quarter of this plane is the image of the exponential map, so the logarithm is only defined on that quarter (quadrant). The other three quadrants are images of this one under the [[Klein four-group]] generated by ε and −1.

For example, let ''a'' = log 2 ; then cosh ''a'' = 5/4 and sinh ''a'' = 3/4.