Greek ladders via linear algebra

Theon's ladder is an ancient method for easily approximating nth roots of a real number k. Previous work in this area has focused on modifying Theon's ladder to approximate roots of quadratic polynomials. We extend this work using techniques from linear algebra. We will show that a ladder associated to the quadratic polynomial ax ² + bx + c can be adjusted to approximate either root. Other situations such as quadratics with no real roots and corresponding matrices with complex eigenvalues are also addressed.