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*If ''A'' and ''B'' are both [[positive-definite matrices]] and if they commute, i.e., ''AB'' = ''BA'', then
:<math>\log{(AB)} = \log{(A)}+\log{(B)}. \, </math>
In fact, for two matrices <math>A, B \in \mathbb{C}^{n \times n}</math> we have <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 J. MATRIX ANAL. APPL. |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}}</ref>.
*Substituting in this equation ''B'' = ''A<sup>−1</sup>'', one gets
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