Exceptional points in two simple textbook examples

TL;DR

This paper introduces the concept of exceptional points through simple textbook examples in mathematics and mechanics, illustrating their connection to defective matrices and Jordan forms in differential equations.

Contribution

It presents accessible examples for teaching exceptional points, linking them to classical differential equations and matrix theory in an educational context.

Findings

Exceptional points relate to linearly dependent solutions.

Defective matrices cannot be diagonalized but can be transformed into Jordan form.

Simple examples help in teaching complex concepts.

Abstract

We propose to introduce the concept of exceptional points in intermediate courses on mathematics and classical mechanics by means of simple textbook examples. The first one is an ordinary second-order differential equation with constant coefficients. The second one is the well known damped harmonic oscillator. They enable one to connect the occurrence of linearly dependent exponential solutions with a defective matrix that cannot be diagonalized but can be transformed into a Jordan canonical form.