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- #1

- Jun 20, 2014

- 1,892

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Define the logarithm of an $n\times n$ matrix $A$ by the power series

$$\sum_{k = 1}^\infty \frac{(-1)^{k-1}(A - I)^k}{k}$$

which converges for $\|A - I\| < 1$ (the standard matrix norm is being used here). Prove that for all $n\times n$ matrices $A$ and $B$ with $\|A - I\| < 1$, $\|B - I\| < 1$, $\|AB - I\| < 1$, and $AB = BA$,

$$\log(AB) = \log A + \log B$$

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