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perhaps a useful identity

21/06/2011

Let X be a positive definite matrix.

\mbox{det} X=\mbox{exp} \mbox{tr} \mbox{ln} X

or

\mbox{ln} \mbox{det} X=\mbox{tr} \mbox{ln }X

See here for application.

 

EDIT: very useful, especially for MLE of Gaussian parameters. In addition,

\mathrm{d}\ln A=A^{-1}\mathrm{d}A

\mathrm{d}\ln |A|=\mathrm{tr}(A^{-1}\mathrm{d}A)A

Because: \mathrm{d}\ln |A|=\mathrm{d}\mathrm{tr}\ln A=\mathrm{tr}(\mathrm{d}\ln A)=\mathrm{tr}A^{-1}\mathrm{d}A

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