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Next: Theorems of Special Interest Up: Continued Fraction Factorization Previous: Motivation and Examples

Matrix Representations and Fractional Linear Transformations

As Justin Miller describes in his report Families of Continued Fractions, there is a fundamental corresponce between continued fractions and matrices. This is a consequence of the fact that if $x=[a_0,a_1,\ldots,a_n]$, then

\begin{displaymath}
\prod_{i=0}^n
\left
(\begin{array}{cc}
0 & 1\\
...
...cc}
Q_{n-1} & Q_n\\
P_{n-1} & P_n
\end{array} \right)
\end{displaymath} (2)


where $\frac{P_i}{Q_i}$ is the ith convergent of the continued fraction expansion of $x$. This correspondence will be the basis for the method of factorization described in this report. Let $\{a_0,a_1,\ldots,a_n\}$ represent the matrix product

\begin{displaymath}
\prod_{i=0}^n
\left
(\begin{array}{cc}
0 & 1\\
...
...begin{array}{cc}
0 & 1\\
1 & a_n
\end{array} \right).
\end{displaymath} (3)


Since we are interested in the string $[a_1,a_2,\ldots,a_n]$ in the expansion $\sqrt{d}=[a_0;\overline{a_1,a_2,\ldots,a_n,b,a_n,\ldots,a_2.a_1,2a_0}]$, it will be convinient to use the following declaration throughout this report:

\begin{displaymath}
M=
\left
(\begin{array}{cc}
A & B\\
C & D
\en...
...begin{array}{cc}
0 & 1\\
1 & a_i
\end{array} \right).
\end{displaymath} (4)


The final tool we will use in this report involves a type of function known as a fractional linear transformation. For a matrix $T=\left
(\begin{array}{cc}
a & b\\
c & d
\end{array} \right)$ and a real number $x$, the fractional linear transformation $T(x)$ is defined as follows:

\begin{displaymath}
T(x)=
\left
(\begin{array}{cc}
a & b\\
c & d
\end{array} \right)(x)=
\frac{ax+b}{cx+d}
\end{displaymath} (5)


We are now ready to introduce and prove certain theorems that are of special interest in regards to this report.


next up previous
Next: Theorems of Special Interest Up: Continued Fraction Factorization Previous: Motivation and Examples

scanez 2000-12-04