Logarithm of a matrix: Difference between revisions

Revision as of 18:03, 13 December 2022

Revision as of 04:31, 22 December 2022

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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=THEThe MATRIXMatrix UNWINDINGUnwinding FUNCTIONFunction, WITHwith ANan APPLICATIONApplication TOto COMPUTINGComputing THEthe MATRIXMatrix EXPONENTIALExponential |journal=SIAM J.Journal MATRIXon ANAL.Matrix APPL.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}}</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>