GeoGebra Visualizations of Eigenvalues, Eigenvectors, and Eigenspaces of Symmetric Matrices

Abstract

This article presents four GeoGebra-based visualizations of symmetric matrices designed to support the teaching and learning of eigenvalues, eigenvectors, and eigenspaces. The examples include two-dimensional and three-dimensional matrices, with both invertible and singular cases, as well as distinct and repeated eigenvalues. Students compare eigenvectors with their images under the matrix transformation and observe stretching, direction reversal, direction preservation, and collapse to the zero vector. The constructions also illustrate the orthogonality of eigenspaces associated with distinct eigenvalues, the connection between a zero eigenvalue and the nullspace, and the geometric meaning of a repeated eigenvalue. In the repeated-eigenvalue example, GeoGebra shows that an entire two-dimensional zero-eigenspace is mapped to the zero vector, rather than only the individual eigenvectors displayed. Classroom prompts and extension activities encourage students to move beyond symbolic calculation and coordinate algebraic, geometric, and visual representations. The visualizations may help students develop a more conceptual understanding of symmetric matrices and their eigenspaces.

Keywords: Symmetric matrices, eigenvalues, eigenvectors, eigenspaces, GeoGebra, linear algebra education, dynamic visualization, repeated eigenvalues

Licensed under CC BY-NC-SA 4.0 · ISSN 2162-3856  ·  North American GeoGebra Journal