Matrix diagonalization and singular value decomposition: Static SageMath and dynamic ChatGPT juxtaposed

Abstract

We investigated some difficulties that students often face when studying linear algebra at the undergraduate level, and identified some common mistakes and difficulties they often encountered when dealing with topics that require algorithmic thinking skills such as matrix factorization. In particular, we focused on (orthogonal) diagonalization and singular value decomposition (SVD). We also offered the possibility of exploring these topics using SageMath, a Python-based free open software computer algebra system (CAS) that has been identified to be useful for assisting many students in the computational process even though its output is static by nature. We then explored dynamic ChatGPT by inquiring the chatbot about the topic, either by asking to provide an example or to solve a problem, that is by constructing an (orthogonal) diagonalization or SVD from a particular matrix. By consolidating essential concepts in linear algebra and improving computational skills through effective practice, mastering these topics would become easier and mistakes could be minimized. Static SageMath, in particular, is a great aid for calculation confirmation and handling tedious computations. Although dynamic ChatGPT is relatively unreliable for solving problems in linear algebra, the mistakes it produces could become a valuable tool for improving critical thinking skills.