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''Reverses'' are unary operations in geometric algebra that are analogs of conjugate or transpose operations.
''Reverses'' are unary operations in geometric algebra that are analogs of conjugate or transpose operations.


For any element $$\mathbf x$$ that is the [[wedge product]] of $$k$$ vectors, the ''reverse'' of $$\mathbf x$$, which we denote by $$\mathbf{\tilde x}$$, is the result of multiplying those same $$k$$ vectors in reverse order. For example, the reverse of $$\mathbf e_{234}$$ is $$\mathbf e_4 \wedge \mathbf e_3 \wedge \mathbf e_2$$, which we would write as $$-\mathbf e_{234}$$since 432 is an odd permutation of 234. In general, the reverse of an element $$\mathbf x$$ is given by
For any element $$\mathbf u$$ that is the [[wedge product]] of $$k$$ vectors, the ''reverse'' of $$\mathbf u$$, which we denote by $$\mathbf{\tilde u}$$, is the result of multiplying those same $$k$$ vectors in reverse order. For example, the reverse of $$\mathbf e_{423}$$ is $$\mathbf e_3 \wedge \mathbf e_2 \wedge \mathbf e_4$$, which we would write as $$-\mathbf e_{423}$$since 324 is an odd permutation of 423. In general, the reverse of an element $$\mathbf u$$ is given by


:$$\mathbf{\tilde x} = (-1)^{\operatorname{gr}(\mathbf x)(\operatorname{gr}(\mathbf x) - 1)/2}\,\mathbf x$$ .
:$$\mathbf{\tilde u} = (-1)^{\operatorname{gr}(\mathbf u)(\operatorname{gr}(\mathbf u) - 1)/2}\,\mathbf u$$ .


Symmetrically, for any element $$\mathbf x$$ that is the [[antiwedge product]] of $$m$$ antivectors, the ''antireverse'' of $$\mathbf x$$, which we denote by $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}}$$, is the result of multiplying those same $$m$$ antivectors in reverse order (but this time under the [[antiwedge product]]). In general, the antireverse of an element $$\mathbf x$$ is given by
Symmetrically, for any element $$\mathbf u$$ that is the [[antiwedge product]] of $$m$$ antivectors, the ''antireverse'' of $$\mathbf u$$, which we denote by $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}}$$, is the result of multiplying those same $$m$$ antivectors in reverse order (but this time under the [[antiwedge product]]). In general, the antireverse of an element $$\mathbf u$$ is given by


:$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}} = (-1)^{\operatorname{ag}(\mathbf x)(\operatorname{ag}(\mathbf x) - 1)/2}\,\mathbf x$$ .
:$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} = (-1)^{\operatorname{ag}(\mathbf u)(\operatorname{ag}(\mathbf u) - 1)/2}\,\mathbf u$$ .


The reverse and antireverse of any element $$\mathbf x$$ are related by
The reverse and antireverse of any element $$\mathbf u$$ are related by


:$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}} = (-1)^{\operatorname{gr}(\mathbf x)\operatorname{ag}(\mathbf x)}(-1)^{n(n-1)/2}\,\mathbf{\tilde x}$$ ,
:$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} = (-1)^{\operatorname{gr}(\mathbf u)\operatorname{ag}(\mathbf u)}(-1)^{n(n-1)/2}\,\mathbf{\tilde u}$$ ,


where $$n$$ is the number of dimensions in the algebra. To extend the reversals to all elements of an algebra, we simply require that it is a linear operation. For any basis elements $$\mathbf x$$ and $$\mathbf y$$, and for any scalars $$a$$ and $$b$$, we must have, for the reverse
where $$n$$ is the number of dimensions in the algebra. To extend the reversals to all elements of an algebra, we simply require that it is a linear operation. For any basis elements $$\mathbf x$$ and $$\mathbf y$$, and for any scalars $$a$$ and $$b$$, we must have, for the reverse
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[[Image:Reverses.svg|720px]]
[[Image:Reverses.svg|720px]]
== In the Book ==
* Reverses and antireverses are introduced in Section 3.4.


== See Also ==
== See Also ==


* [[Complements]]
* [[Complements]]

Latest revision as of 23:32, 13 April 2024

Reverses are unary operations in geometric algebra that are analogs of conjugate or transpose operations.

For any element $$\mathbf u$$ that is the wedge product of $$k$$ vectors, the reverse of $$\mathbf u$$, which we denote by $$\mathbf{\tilde u}$$, is the result of multiplying those same $$k$$ vectors in reverse order. For example, the reverse of $$\mathbf e_{423}$$ is $$\mathbf e_3 \wedge \mathbf e_2 \wedge \mathbf e_4$$, which we would write as $$-\mathbf e_{423}$$since 324 is an odd permutation of 423. In general, the reverse of an element $$\mathbf u$$ is given by

$$\mathbf{\tilde u} = (-1)^{\operatorname{gr}(\mathbf u)(\operatorname{gr}(\mathbf u) - 1)/2}\,\mathbf u$$ .

Symmetrically, for any element $$\mathbf u$$ that is the antiwedge product of $$m$$ antivectors, the antireverse of $$\mathbf u$$, which we denote by $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}}$$, is the result of multiplying those same $$m$$ antivectors in reverse order (but this time under the antiwedge product). In general, the antireverse of an element $$\mathbf u$$ is given by

$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} = (-1)^{\operatorname{ag}(\mathbf u)(\operatorname{ag}(\mathbf u) - 1)/2}\,\mathbf u$$ .

The reverse and antireverse of any element $$\mathbf u$$ are related by

$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} = (-1)^{\operatorname{gr}(\mathbf u)\operatorname{ag}(\mathbf u)}(-1)^{n(n-1)/2}\,\mathbf{\tilde u}$$ ,

where $$n$$ is the number of dimensions in the algebra. To extend the reversals to all elements of an algebra, we simply require that it is a linear operation. For any basis elements $$\mathbf x$$ and $$\mathbf y$$, and for any scalars $$a$$ and $$b$$, we must have, for the reverse

$$\widetilde{(a\mathbf x + b\mathbf y)} = a\mathbf{\tilde x} + b\mathbf{\tilde y}$$ ,

and similarly for the antireverse.

The following table lists the reverse and antireverse for all of the basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.

In the Book

  • Reverses and antireverses are introduced in Section 3.4.

See Also