In geometric algebra, the outermorphism of a linear function between vector spaces is a natural extension of the map to arbitrary multivectors.^[1] It is the unique unital algebra homomorphism of exterior algebras whose restriction to the vector spaces is the original function.^[a]

Definition

Let ${\displaystyle f}$ be an ${\displaystyle \mathbb {R} }$-linear map from ${\displaystyle V}$ to ${\displaystyle W}$. The extension of ${\displaystyle f}$ to an outermorphism is the unique map ${\displaystyle \textstyle {\underline {\mathsf {f}}}:\bigwedge (V)\to \bigwedge (W)}$ satisfying

${\displaystyle {\underline {\mathsf {f}}}(1)=1}$

${\displaystyle {\underline {\mathsf {f}}}(x)=f(x)}$

${\displaystyle {\underline {\mathsf {f}}}(A\wedge B)={\underline {\mathsf {f}}}(A)\wedge {\underline {\mathsf {f}}}(B)}$

${\displaystyle {\underline {\mathsf {f}}}(A+B)={\underline {\mathsf {f}}}(A)+{\underline {\mathsf {f}}}(B)}$

for all vectors ${\displaystyle x}$ and all multivectors ${\displaystyle A}$ and ${\displaystyle B}$, where ${\displaystyle \textstyle \bigwedge (V)}$ denotes the exterior algebra over ${\displaystyle V}$. That is, an outermorphism is a unital algebra homomorphism between exterior algebras.

The outermorphism inherits linearity properties of the original linear map. For example, we see that for scalars ${\displaystyle \alpha }$, ${\displaystyle \beta }$ and vectors ${\displaystyle x}$, ${\displaystyle y}$, ${\displaystyle z}$, the outermorphism is linear over bivectors:

${\displaystyle {\begin{aligned}{\underline {\mathsf {f}}}(\alpha x\wedge z+\beta y\wedge z)&={\underline {\mathsf {f}}}((\alpha x+\beta y)\wedge z)\\&=f(\alpha x+\beta y)\wedge f(z)\\&=(\alpha f(x)+\beta f(y))\wedge f(z)\\&=\alpha (f(x)\wedge f(z))+\beta (f(y)\wedge f(z))\\&=\alpha \,{\underline {\mathsf {f}}}(x\wedge z)+\beta \,{\underline {\mathsf {f}}}(y\wedge z),\end{aligned}}}$

which extends through the axiom of distributivity over addition above to linearity over all multivectors.

Adjoint

Let ${\displaystyle {\underline {\mathsf {f}}}}$ be an outermorphism. We define the adjoint of ${\displaystyle {\overline {\mathsf {f}}}}$ to be the outermorphism that satisfies the property

${\displaystyle {\overline {\mathsf {f}}}(a)\cdot b=a\cdot {\underline {\mathsf {f}}}(b)}$

for all vectors ${\displaystyle a}$ and ${\displaystyle b}$, where ${\displaystyle \cdot }$ is the nondegenerate symmetric bilinear form (scalar product of vectors).

This results in the property that

${\displaystyle {\overline {\mathsf {f}}}(A)*B=A*{\underline {\mathsf {f}}}(B)}$

for all multivectors ${\displaystyle A}$ and ${\displaystyle B}$, where ${\displaystyle *}$ is the scalar product of multivectors.

If geometric calculus is available, then the adjoint may be extracted more directly:

${\displaystyle {\overline {\mathsf {f}}}(a)=\nabla _{b}\left\langle a{\underline {\mathsf {f}}}(b)\right\rangle .}$

The above definition of adjoint is like the definition of the transpose in matrix theory. When the context is clear, the underline below the function is often omitted.

Properties

It follows from the definition at the beginning that the outermorphism of a multivector ${\displaystyle A}$ is grade-preserving:^[2]

${\displaystyle {\underline {\mathsf {f}}}(\left\langle A\right\rangle _{r})=\left\langle {\underline {\mathsf {f}}}(A)\right\rangle _{r}}$

where the notation ${\displaystyle \langle ~\rangle _{r}}$ indicates the ${\displaystyle r}$-vector part of ${\displaystyle A}$.

Since any vector ${\displaystyle x}$ may be written as ${\displaystyle }$Zdroj:https://en.wikipedia.org?pojem=Outermorphism
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