 Continued-fraction matrix

A continued-fraction matrix packages the recurrences for continued fraction convergents into matrix multiplication. PutIf $p_n/q_n=[a_0;\dots ,a_n]$, with $p_{-2}=0$, $p_{-1}=1$, $q_{-2}=1$ and $q_{-1}=0$, thenFor $n=0$ the identity is immediate. Right multiplication by $M(a_n)$ replaces the first column by $a_n$ times the old first column plus the old second column, proving the convergent recurrences by mathematical induction.

Each factor has determinant $-1$, so $p_n{q}_{n-1}-p_{n-1}{q}_n=(-1{)}^{n+1}$. Any common divisor of $p_n$ and $q_n$ must therefore divide $1$. Thus each continued fraction convergent is a fraction with coprime numerator and denominator.