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PostPosted: Sat Mar 11, 2017 8:47 pm 

Joined: Sat Nov 14, 2015 6:32 am
Posts: 146
Location: Melbourne, Australia
Let $A$ be a $2 \times 2$ matrix with rational entries and both eigenvalues less than one in absolute value. Prove that

$$\log \left( \mathbb{I}_{2 \times 2} - A \right) = - \sum_{n=1}^{\infty} \frac{A^n}{n}$$

has rational entries if and only if $A^2 = \mathbb{O}_{2 \times 2}$.

_________________
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$


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