Continued Fraction Expansions of Matrix Eigenvectors

TL;DR

This paper investigates the properties of continued fraction expansions of eigenvector slopes of SL(2, Z) matrices, providing statistical analyses and proving palindromic period properties for certain algebraic roots.

Contribution

It introduces new statistical measures for continued fraction periods of matrix eigenvectors and proves palindromic period properties for roots of quadratic equations.

Findings

Average and maximum period lengths calculated

Distribution patterns of partial quotients analyzed

Palindromic nature of periods for quadratic roots proved

Abstract

We examine various properties of the continued fraction expansions of matrix eigenvector slopes of matrices from the SL(2, Z) group. We calculate the average period length, maximum period length, average period sum, maximum period sum and the distributions of 1s 2s and 3s in the periods versus the radius of the Ball within which the matrices are located. We also prove that the periods of continued fraction expansions from the real irrational roots of x2 + px + q = 0 are always palindromes.