Magnitude and Reverses: Difference between pages
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''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 | |||
:$$\mathbf{\tilde x} = (-1)^{\operatorname{gr}(\mathbf x)(\operatorname{gr}(\mathbf x) - 1)/2}\,\mathbf x$$ . | |||
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 | |||
:$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}} = (-1)^{\operatorname{ag}(\mathbf x)(\operatorname{ag}(\mathbf x) - 1)/2}\,\mathbf x$$ . | |||
The reverse and antireverse of any element $$\mathbf x$$ are related by | |||
:$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}} = (-1)^{\operatorname{gr}(\mathbf x)\operatorname{ag}(\mathbf x)}(-1)^{n(n-1)/2}\,\mathbf{\tilde x}$$ , | |||
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 $$\mathcal G_{3,0,1}$$. | |||
[[Image:Reverses.svg|720px]] | |||
== See Also == | == See Also == | ||
* [[ | * [[Complements]] | ||
Revision as of 08:01, 22 January 2024
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
- $$\mathbf{\tilde x} = (-1)^{\operatorname{gr}(\mathbf x)(\operatorname{gr}(\mathbf x) - 1)/2}\,\mathbf x$$ .
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
- $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}} = (-1)^{\operatorname{ag}(\mathbf x)(\operatorname{ag}(\mathbf x) - 1)/2}\,\mathbf x$$ .
The reverse and antireverse of any element $$\mathbf x$$ are related by
- $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}} = (-1)^{\operatorname{gr}(\mathbf x)\operatorname{ag}(\mathbf x)}(-1)^{n(n-1)/2}\,\mathbf{\tilde x}$$ ,
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 $$\mathcal G_{3,0,1}$$.