Two eigenvectors for the price of one

TL;DR

The paper presents a novel method to compute both eigenvectors of a 2x2 matrix with distinct eigenvalues simultaneously, connecting it to the Cayley-Hamilton theorem and extending it to 3x3 matrices.

Contribution

It introduces an unexpected, simple technique for finding both eigenvectors at once and generalizes the approach to larger matrices, enhancing linear algebra methods.

Findings

Method finds both eigenvectors simultaneously for 2x2 matrices.

Connection established with Cayley-Hamilton theorem.

Generalization demonstrated for 3x3 matrices.

Abstract

Starting from a mistake done by a student, we discover an unexpected method of finding both eigenvectors for a $2\times2$ matrix with distinct eigenvalues in a single computation. We discuss a connection with the Cayley-Hamilton theorem, and show the corresponding generalization for a $3 \times 3$ matrix. The arguments should be understandable for strong linear algebra students.