The Fundamental Correspondence

where

For the remainder of the report, let denote the matrix product

At first, this correspondence between continued fractions and matrices might not seem to be very useful, but when it is used in conjunction with fractional linear transformations, it becomes a powerful tool for the continued fractions of surds. Let denote the fractional linear transformation associated with , where

In general, the relationship between continued fractions and fractional linear transformations is the following.

Now that this theorem has been established, it is possible to prove a proposition that is of prime importance in the the study of the continued fractions of pure quadratic irrationals.