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All Rights Reserved. All the content available on this site is protected by copyright.
All trademarks are the property of their respective owners. as an equivalence relation (hence a congruence relation).
This operation allows us to generalize the concept of Clifford numbers. Indeed,
is also a relation on.
From this equivalence we obtain an involution on the Clifford algebra. Moreover, we can use this to compute the index of for a real matrix,
where is the supermetric and is the matrix representation of a linear transformation.
When has dimension and has dimension, we have an orthogonal projection from the complex vector space to. If is the natural inner product in the vector space, then the matrix can be identified with the orthogonal projection. Then is the Clifford product, and so
where is the matrix representation of the orthogonal projection. In general,
where is a Euclidean space,, and is the Clifford product.
The Clifford group is the group of unitary transformations that preserve the inner product and the Clifford product.
The distinction between a matrix and its transpose is now trivial: a transpose is the matrix obtained from a given matrix by exchanging the rows and columns, and using conjugation.
See also
Adjoint representation
Clifford algebra
Lie algebra
Notes
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