Maths - Clifford Algebra - Quadratic Form

Maths - Quadratic Form

The inner product defines the type of Clifford algebra. For instance which dimensions square to positive, which to negative and so on.

This can be defined by the quadratic form (for introduction to quadratic form see this page).

This inner product is important, for instance, the way that a vector square to a scalar number represents the metric (the way that we define distance) of a space.

Bilinear and Quadratic Forms

A quadratic form is an expression in a number of variables where each term is of degree two.

Every quadratic form can be represented by a symmetric matrix, to generate the quadratic form we pre-multiply the matrix by a single row vector containing the variables and post-multiply the matrix by a single column vector also containing the variables.

1	3
3	1

=x² + 6xy + y²

More about bilinear form on this page and quadratic form on this page.

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see also:

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Fundamental Theories of Physics). This book is intended for mathematicians and physicists rather than programmers, it is very theoretical. It covers the algebra and calculus of multivectors of any dimension and is not specific to 3D modelling.

New Foundations for Classical Mechanics (Fundamental Theories of Physics). This is very good on the geometric interpretation of this algebra. It has lots of insights into the mechanics of solid bodies. I still cant work out if the position, velocity, etc. of solid bodies can be represented by a 3D multivector or if 4 or 5D multivectors are required to represent translation and rotation.

Terminology and Notation

Specific to this page here:

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Maths - Quadratic Form

Bilinear and Quadratic Forms

EuclideanSpace

Category Theory