Reverses

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

Every basis element $$\mathbf a$$ has a reverse, which we denote by $$\mathbf{\tilde a}$$, that satisfies the equation

$$\mathbf a \mathbin{\unicode{x27D1}} \mathbf{\tilde a} = \mathbf 1$$ .

There is also an antireverse, which we denote by $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{a}}}$$, that satisfies the equation

$$\mathbf a \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{a}}} = {\large\unicode{x1D7D9}}$$ .

For any element $$\mathbf a$$ that is the wedge product of $$k$$ vectors, the reverse of $$\mathbf a$$ 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 $$\mathbf a$$ is given by

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

Symmetrically, for any element $$\mathbf a$$ that is the antiwedge product of $$m$$ antivectors, the antireverse of $$\mathbf a$$ 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 $$\mathbf a$$ is given by

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

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

$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{a}}} = (-1)^{\operatorname{gr}(\mathbf a)\operatorname{ag}(\mathbf a)}\,\mathbf{\tilde a}$$ .

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

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

and similarly for the antireverse.

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

Element $$\mathbf a$$ $$\mathbf 1$$ $$\mathbf e_1$$ $$\mathbf e_2$$ $$\mathbf e_3$$ $$\mathbf e_4$$ $$\mathbf e_{23}$$ $$\mathbf e_{31}$$ $$\mathbf e_{12}$$ $$\mathbf e_{43}$$ $$\mathbf e_{42}$$ $$\mathbf e_{41}$$ $$\mathbf e_{321}$$ $$\mathbf e_{124}$$ $$\mathbf e_{314}$$ $$\mathbf e_{234}$$ $$\large\unicode{x1D7D9}$$
Reverse $$\mathbf{\tilde a}$$ $$\mathbf 1$$ $$\mathbf e_1$$ $$\mathbf e_2$$ $$\mathbf e_3$$ $$\mathbf e_4$$ $$-\mathbf e_{23}$$ $$-\mathbf e_{31}$$ $$-\mathbf e_{12}$$ $$-\mathbf e_{43}$$ $$-\mathbf e_{42}$$ $$-\mathbf e_{41}$$ $$-\mathbf e_{321}$$ $$-\mathbf e_{124}$$ $$-\mathbf e_{314}$$ $$-\mathbf e_{234}$$ $$\large\unicode{x1D7D9}$$
Antireverse $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{a}}}$$ $$\mathbf 1$$ $$-\mathbf e_1$$ $$-\mathbf e_2$$ $$-\mathbf e_3$$ $$-\mathbf e_4$$ $$-\mathbf e_{23}$$ $$-\mathbf e_{31}$$ $$-\mathbf e_{12}$$ $$-\mathbf e_{43}$$ $$-\mathbf e_{42}$$ $$-\mathbf e_{41}$$ $$\mathbf e_{321}$$ $$\mathbf e_{124}$$ $$\mathbf e_{314}$$ $$\mathbf e_{234}$$ $$\large\unicode{x1D7D9}$$

See Also