Complements and Duals: Difference between pages

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''Complements'' are unary operations in geometric algebra that perform a specific type of dualization.
Every object in projective geometric algebra has two duals derived from the metric tensor, called the ''metric dual'' and ''metric antidual''.


Every basis element $$\mathbf u$$ has a ''right complement'', which we denote by $$\overline{\mathbf u}$$, that satisfies the equation
== Dual ==


:$$\mathbf u \wedge \overline{\mathbf u} = {\large\unicode{x1D7D9}}$$ .
The ''metric dual'' or just "dual" of an object $$\mathbf u$$ is denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2605}$$ and defined as


There is also a ''left complement'', which we denote by $$\underline{\mathbf u}$$, that satisfies the equation
:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf{Gu}}$$ ,


:$$\underline{\mathbf u} \wedge \mathbf u = {\large\unicode{x1D7D9}}$$ .
where $$\mathbf G$$ is the $$16 \times 16$$ metric exomorphism matrix. In projective geometric algebra, this dual is also called the ''bulk dual'' because it is the [[complement]] of the bulk components, as expressed by


Complements exchange full and empty dimensions, and the left and right complements can differ only by sign according to the relationship
:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf u_\unicode{x25CF}}$$ .


:$$\underline{\mathbf u} = (-1)^{\operatorname{gr}(\mathbf u)\operatorname{ag}(\mathbf u)}\,\overline{\mathbf u}$$ .
The bulk dual satisfies the following identity based on the [[geometric product]]:


This shows that the left and right complements of an element $$\mathbf u$$ are always the same if either its [[grade]] $$\operatorname{gr}(\mathbf u)$$ or its [[antigrade]] $$\operatorname{ag}(\mathbf u)$$ is even. If the number of dimensions is odd, then it is always true that one of these is even, so left and right complements are the same for all elements in an odd-dimensional algebra. As shown in the table below, applying the right or left complement twice can negate the operand in even numbers of dimensions. However, the right and left complements are inverse operations, so we always have $$\overline{\underline{\mathbf u}} = \mathbf u$$.
:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \tilde{\mathbf u} \mathbin{\unicode{x27D1}} {\large\unicode{x1D7D9}}$$ .


Taking the right or left complement twice causes the sign to change according to the formula
== Antidual ==


:$$\underline{\underline{\mathbf u}} = \overline{\overline{\mathbf u}} = (-1)^{\operatorname{gr}(\mathbf u)\operatorname{ag}(\mathbf u)}\,\mathbf u$$ .
The ''metric antidual'' or just "antidual" of an object $$\mathbf u$$ is denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2606}$$ and defined as


The right and left complements under the [[wedge product]] are also the right and left complements under the [[antiwedge product]], so we can write
:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbb G \mathbf u}$$ ,


:$$\mathbf u \vee \overline{\mathbf u} = \mathbf 1$$
where $$\mathbb G$$ is the $$16 \times 16$$ metric antiexomorphism matrix. In projective geometric algebra, this dual is also called the ''weight dual'' because it is the [[complement]] of the weight components, as expressed by
:$$\underline{\mathbf u} \vee\mathbf u = \mathbf 1$$ .


To extend the complements to all elements of an algebra, we simply require that they are linear operations. For any basis elements $$\mathbf x$$ and $$\mathbf y$$, and for any scalars $$a$$ and $$b$$, we must have, for the right complement,
:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbf u_\unicode{x25CB}}$$ .


:$$\overline{(a\mathbf x + b\mathbf y)} = a\overline{\mathbf x} + b\overline{\mathbf y}$$ ,
The weight dual satisfies the following identity based on the [[geometric antiproduct]]:


and similarly for the left complement.
:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} \mathbin{\unicode{x27C7}} \mathbf 1$$ .
 
The following table lists the left and right complements for all of the basis elements in the 4D geometric algebra $$\mathcal G_{3,0,1}$$.
 
[[Image:Complements.svg|720px]]


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


* [[Duals]]
* [[Complements]]
* [[Grade and antigrade]]
* [[Bulk and weight]]
* [[Bulk and weight]]
* [[Reverses]]
* [[Duality]]

Revision as of 01:36, 13 April 2024

Every object in projective geometric algebra has two duals derived from the metric tensor, called the metric dual and metric antidual.

Dual

The metric dual or just "dual" of an object $$\mathbf u$$ is denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2605}$$ and defined as

$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf{Gu}}$$ ,

where $$\mathbf G$$ is the $$16 \times 16$$ metric exomorphism matrix. In projective geometric algebra, this dual is also called the bulk dual because it is the complement of the bulk components, as expressed by

$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf u_\unicode{x25CF}}$$ .

The bulk dual satisfies the following identity based on the geometric product:

$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \tilde{\mathbf u} \mathbin{\unicode{x27D1}} {\large\unicode{x1D7D9}}$$ .

Antidual

The metric antidual or just "antidual" of an object $$\mathbf u$$ is denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2606}$$ and defined as

$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbb G \mathbf u}$$ ,

where $$\mathbb G$$ is the $$16 \times 16$$ metric antiexomorphism matrix. In projective geometric algebra, this dual is also called the weight dual because it is the complement of the weight components, as expressed by

$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbf u_\unicode{x25CB}}$$ .

The weight dual satisfies the following identity based on the geometric antiproduct:

$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} \mathbin{\unicode{x27C7}} \mathbf 1$$ .

See Also

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