1.5.3 The Matrix Logarithm

We know that the matrix ex­po­nen­tial of a matrix $M$ is well-de­fined and ab­solute­ly con­ver­gent

for which, the matrix log­a­rithm rep­re­sents the in­verse func­tion for the ex­po­nen­tial map­ping $\exp^{- 1}(M)$. The matrix log­a­rithm, like the clas­si­cal log, is de­fined by a power series that only con­verges in a cer­tain neigh­bor­hood of the iden­ti­ty. To start, it is im­por­tant to first prove that $(1 + u)^{- 1}$ equals the fol­low­ing in­fi­nite series

Set­ting $S$ equal to the above in­fi­nite series

we sub­tract one from each side then divide both sides by $u$

which re­sults in a re­la­tion­ship in­volv­ing $S$ and $u$, equal to the ini­tial in­fi­nite series

We divide both sides of the above by $S$, then mul­ti­ply­ing through by $u$

which can be sim­pli­fied and solved for $S$, to com­plete the proof

$\square$

It turns out that we can de­scribe the log­a­rithm of $x$ as a re­la­tion­ship of this in­fi­nite series. For $|x - 1| \leq 1$, and $x \neq 0$, the log­a­rithm can be ex­pressed as the in­te­gral

where chang­ing the bounds of in­te­gra­tion and making a $u$ sub­sti­tu­tion, the in­te­gral can be ex­pressed, as

whose in­te­grand equals the in­fi­nite series of the proof

Solv­ing this in­te­gral is straight­for­ward since it is in the form of a poly­no­mi­al, which can be ex­pressed as the ex­pand­ed in­fi­nite sum

which is equiv­a­lent to the Taylor series for $\ln(x)$, around the iden­ti­ty. A change of vari­ables re­sults in the Mer­ca­tor series

valid for $|x| \leq 1$, and $x \neq - 1$. This clas­si­cal in­fi­nite series can be ex­tend­ed to ma­tri­ces, with which we can define the log­a­rithm of a square matrix, as

for $\epsilon$, the iden­ti­ty matrix. The func­tion $\ln(\epsilon + A)$, is well-de­fined and con­tin­u­ous in the neigh­bor­hood of the iden­ti­ty $\epsilon$, and ab­solute­ly con­ver­gent for $|A| < 1$.

The­o­rem (1.5.31) – The in­verse prop­er­ty of the matrix log­a­rithm will hold for any matrix $e^{X}$, within the unit dis­tance of the iden­ti­ty, as de­scribed by Still­well ¹

Proof - As $e^{X} = \epsilon + \frac{X}{1!} + \frac{X^{2}}{2!} + \frac{X^{3}}{3!} + \frac{X^{4}}{4!} + \ldots,$ and $\left| e^{X} - 1 \right| < 1$, we have

by way of the matrix log­a­rithm de­f­i­n­i­tion and the Mer­ca­tor series. As the series is ab­solute­ly con­ver­gent, the terms can be arranged by powers of $X^{m}$

The higher-pow­ered co­ef­fi­cients of $X^{m}$ are zero

and we can say with cer­tain­ty that all the en­su­ing co­ef­fi­cients in the series will sum to zero since the same terms occur in the ex­pres­sion $\ln\left( e^{x} \right)$, for $\left| e^{x} - 1 \right| < 1,$ with sums equal to zero since $\ln\left( e^{x} \right) = x$, under these con­di­tions. $\therefore\ln\left( e^{X} \right) = X.$

$\square$

1. Still­well, John, "Naive Lie Theory", Springer 2008