Logarithm of a matrix: Difference between revisions

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== Power series expression ==

If ''B'' is sufficiently close to the identity matrix, then a logarithm of ''B'' may be computed by means of the [[power series]]

: <math>\log(B) = \log(I + (B - I)) = \sum_{k=1}^{\infty} \frac{(-1)^{k + 1}\frac}{k} (B - I)^k}{k}} = (B - I) - \frac{(B - I)^2}{2} + \frac{(B - I)^3}{3} -\frac{(B-I)^4}{4}+ \cdots</math>.,

which can be rewritten as

Specifically, if <math>\left\|B-I\right\|<1</math>, then the preceding series converges and <math>e^{\log(B)}=B</math>.<ref>{{harvnb|Hall|2015}} Theorem 2.8</ref>

:<math>\log(B) = -\sum_{k=1}^{\infty} \frac{(I - B)^k}{k} = -(I - B) - \frac{(I - B)^2}{2} - \frac{(I - B)^3}{3} - \cdots</math>.

Specifically, if <math>\left\|B-I-B\right\|<1</math>, then the preceding series converges and <math>e^{\log(B)}=B</math>.<ref>{{harvnb|Hall|2015}} Theorem 2.8</ref>

== Example: Logarithm of rotations in the plane ==

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<math>

\sum_{k=0}^\infty{1 \over k!}B_n^k =\begin{pmatrix}

\sum_{k=0}^\infty{(-1)^k \over 2k!}(\alpha+2\pi n)^{2k} & -\sum_{k=0}^\infty{(-1)^k \over (2k+1)!}(\alpha+2\pi n)^{2k+1} \\

\sum_{k=0}^\infty{(-1)^k \over (2k+1)!}(\alpha+2\pi n)^{2k+1} & \sum_{k=0}^\infty{(-1)^k \over 2k!}(\alpha+2\pi n)^{2k} \\

\end{pmatrix} =\begin{pmatrix}

\cos(\alpha) & -\sin(\alpha) \\

\sin(\alpha) & \cos(\alpha) \\

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Suppose that ''A'' and ''B'' commute, meaning that ''AB'' = ''BA''. Then

: <math>\log{(AB)} = \log{(A)}+\log{(B)}</math>

if and only if <math>\operatorname{arg}(\mu_j) + \operatorname{arg}(\nu_j) \in (- \pi, \pi]</math>, where <math>\mu_j</math> is an [[eigenvalue]] of <math>A</math> and <math>\nu_j</math> is the corresponding [[eigenvalue]] of <math>B</math>.<ref>{{cite journal |last1=APRAHAMIAN |first1=MARY |last2=HIGHAM |first2=NICHOLAS J. |title=The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential |journal=SIAM Journal on Matrix Analysis and Applications |year=2014 |volume=35 |issue=1 |page=97 |doi=10.1137/130920137 |url=https://epubs.siam.org/doi/pdf/10.1137/130920137 |access-date=13 December 2022|doi-access=free }}</ref> In particular, <math>\log(AB) = \log(A) + \log(B)</math> when ''A'' and ''B'' commute and are both [[Definite matrix|positive-definite]]. Setting ''B'' = ''A⁻¹'' in this equation yields

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

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== Constraints in the 2 × 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 ''''ε² ∈ {{mset| −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 ε² =+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.

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