Chapter 1: Introduction to the Clifford Product

The Clifford product serves as a fundamental operation in Grassmann algebra. It describes the inner product between two vectors, which can be illustrated through the following components:

and

Given the basis vectors e1 and e2, the inner product can be represented as

This signifies the product of the magnitude of vector x and the projection of vector y onto x. The notation

denotes the L2-norm, while ? represents the angle between vectors x and y within their shared plane.

In contrast, the outer product—often referred to as the Wedge product—is defined as

This operation represents the multiplication of vector x with the projection of vector y onto the direction orthogonal to x. The unit bivector

indicates the orientation of the hyperplane formed by

As detailed in Section II of geometric-algebra adaptive filters, the Clifford product (also known as the geometric product, represented by the dot symbol) is expressed as

This algebra is crucial for modeling vector fields, proving essential in applications such as wind velocity analysis and fluid dynamics, notably in the Navier-Stokes equation.

The video "Spinors for Beginners 11: What is a Clifford Algebra? (and Geometric, Grassmann, Exterior Algebras)" provides an insightful overview of Clifford algebras and their geometric interpretations.

In the "Clifford Algebra" video, various aspects of Clifford algebra are discussed, offering a comprehensive introduction to its principles and applications.

References:

• Spinors for Beginners 11: What is a Clifford Algebra? (and Geometric, Grassmann, Exterior Algebras). YouTube.
• A Swift Introduction to Geometric Algebra. YouTube.
• Learning on Graphs & Geometry. Weekly reading groups every Monday at 11 am ET.
• What’s the Clifford algebra? Mathematics Stack Exchange.
• Introducing CliffordLayers: Neural Network layers inspired by Clifford / Geometric Algebras. Microsoft Research AI4Science.
• David Ruhe, Jayesh K. Gupta, Steven de Keninck, Max Welling, Johannes Brandstetter (2023). Geometric Clifford Algebra Networks. arXiv:2302.06594.
• Maksim Zhdanov, David Ruhe, Maurice Weiler, Ana Lucic, Johannes Brandstetter, Patrick Forre (2024). Clifford-Steerable Convolutional Neural Networks. arXiv:2402.14730.