Continued fractions as matrix products

A continued fraction of the form

\cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \ddots}}}

with n terms can be written as the composition

f_1 \circ f_2 \circ f_3 \circ \cdots \circ f_n


f_i(z) = \frac{a_1}{b_i + z}

As discussed in the previous post, a Möbius transformation can be associated with a matrix. And the composition of Möbius transformations is associated with the product of corresponding matrices. So the continued fraction at the top of the post is associated with the following product of matrices.

\begin{pmatrix} 0 & a_1 \\ 1 & b_1\end{pmatrix} \begin{pmatrix} 0 & a_2 \\ 1 & b_2\end{pmatrix} \begin{pmatrix} 0 & a_3 \\ 1 & b_3\end{pmatrix} \cdots \begin{pmatrix} 0 & a_n \\ 1 & b_n\end{pmatrix}

The previous post makes precise the terms “associated with” above: Möbius transformations on the complex plane ℂ correspond to linear transformations on the projective plane P(ℂ). This allows us to include ∞ in the domain and range without resorting to hand waving.

Matrix products are easier to understand than continued fractions, and so moving to the matrix product representation makes it easier to prove theorems.

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