The Matrix Exponential and Matrix Lie Groups

Lie Groups with Applications – Quantum Formalism Series

This interactive Jupyter notebook was created as a supplement to Lecture 1 of the Lie Groups with Applications course, produced by Quantum Formalism in partnership with Zaiku Group. The focus of this first lecture is the matrix exponential, a foundational tool in the study of Lie groups, with deep connections to linear algebra, differential equations, and symmetry.

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🌟 Key Concepts

• What is a matrix Lie group?
• The matrix exponential as a power series: convergence, properties, and geometric meaning
• Exponential maps as solutions to linear ODEs
• Examples from classical groups: $GL(n), SL(n), O(n), SO(n)$
• Diagonalization, Jordan-Chevalley decomposition, and numerical computation using scipy.linalg.expm

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🧑‍💻 What's in the Notebook?

• Numerical and symbolic demonstrations of $e^X$ for real and complex matrices
• Visualizations of Lie group actions and structure-preserving flows
• Worked examples tied to core theorems from the lecture
• Interactive cells for exploring your own matrices and their exponentials

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🎓 Audience & Prerequisites

This material is designed for:

• Learners new to Lie theory
• Students of linear algebra, differential equations, or quantum mechanics
• Researchers and developers in machine learning, control theory, or physics

A working knowledge of basic matrix algebra and calculus is assumed. No prior background in Lie groups is required.

📚 About the Series

This notebook is part of a growing educational series on Lie Groups and Their Applications, produced by Quantum Formalism, whose mission is to make abstract mathematics accessible, visual, and empowering.

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👤 Author

Brian Hepler, PhD
Mathematician, instructor, and consultant in geometry, topology, and AI
LinkedIn · Website · GitHub