On rational entries of a matrix

Linear Algebra
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Riemann
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Location: Melbourne, Australia

On rational entries of a matrix

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Post by Riemann »

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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