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.
- 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
- 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
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.
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.
Brian Hepler, PhD
Mathematician, instructor, and consultant in geometry, topology, and AI
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