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==Properties==
*If ''A'' and ''B'' are both [[positive-definite matrices]], then
:<math>\operatorname{tr}{\log{(AB)}} = \operatorname{tr}{\log{(A)}} + \operatorname{tr}{\log{(B)}}, </math>
*and if ''A'' and ''B'' commute, i.e., ''AB'' = ''BA'', then
:<math>\log{(AB)} = \log{(A)}+\log{(B)}. \, </math>
*Substituting in this equation ''B'' = ''A<sup>−1</sup>'', one gets
:<math> \log{(A^{-1})} = -\log{(A)}.</math>
*Similarly, now for non-commuting ''A'' and ''B'',<ref>[https://www.ias.edu/sites/default/files/sns/files/1-matrixlog_tex(1).pdf Unpublished memo] by S Adler (IAS) </ref>
:<math>\log{(A+tB)} = \log{(A)}+t\int_0^\infty \!\! \! dz ~~\frac{I}{A+zI} B \frac{I}{A+zI} +O(t^2).</math>
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