# Matrix Series : Trace Log and Log Det

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Let K be an nxn matrix and a small number. Imitating

.....

valid for small x, it is natural to define

....

Explain why this makes sense. Prove

trace log = log det(I + K)

Still with small so that everything makes sense.

Hint: What is I − K +2K2 − 3K3 + · · · ?

Since this is an analysis problem, please be sure to be rigorous, and include as much detail as possible so that I can understand. Please also state if you are making use of some fact or theorem. Thanks!

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A matrix series problem is solved.

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