Outermorphism

This article is about outermorphisms of Clifford algebras over a finite-dimensional quadratic space. For the concept in group theory, see outer automorphism group.

In geometric algebra, the outermorphism of a linear function between vectors is a natural extension of the map to arbitrary multivectors.^[1]

Definition

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

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

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

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

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

for all vectors $x$ and all multivectors $A$ and $B$ , where $\bigwedge (V)$ denotes the exterior algebra over $V$ .

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

{\begin{aligned}\underline {{\mathsf {f}}}(\alpha x\wedge z+\beta y\wedge z)&=\underline {{\mathsf {f}}}((\alpha x+\beta y)\wedge z)\\[6pt]&=f(\alpha x+\beta y)\wedge f(z)\\[6pt]&=(\alpha f(x)+\beta f(y))\wedge f(z)\\[6pt]&=\alpha (f(x)\wedge f(z))+\beta (f(y)\wedge f(z))\\[6pt]&=\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 $a$ and $b$ be vectors and $\underline {{\mathsf {f}}}$ be an outermorphism. We define the adjoint function $\overline {{\mathsf {f}}}$ to be the function that satisfies the property

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

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

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

Note that the above definition of adjoint is like the definition of the transpose in matrix theory. The adjoint is itself an outermorphism. 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 $A$ is grade-preserving:^[2]

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

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

Since any vector $x$ may be written as $x=1\wedge x$ , it follows that scalars are unaffected with ${\underline {\mathsf {f}}}(1)=1$ .^[3] Similarly, since there is only ever one independent pseudoscalar, we must have ${\mathsf {f}}(I)\propto I$ . The determinant is defined to be the proportionality factor:^[4]

\det {\mathsf {f}}=\underline {{\mathsf {f}}}(I)I^{{-1}}

The underline is not necessary in this context because the determinant of a function is the same as the determinant of its adjoint. The determinant of the composition of functions is the product of the determinants:

\det({\mathsf {f}}\circ {\mathsf {g}})=\det {\mathsf {f}}\det {\mathsf {g}}

If the determinant of a function is nonzero, then the function has an inverse given by

\underline {{\mathsf {f}}}^{{-1}}(X)={\frac {\overline {{\mathsf {f}}}(XI)I^{{-1}}}{\det {\mathsf {f}}}}=\overline {{\mathsf f}}(XI)[\overline {{\mathsf f}}(I)]^{{-1}},

and so does its adjoint, with

{\overline {\mathsf {f}}}^{-1}(X)={\frac {I^{-1}{\underline {\mathsf {f}}}(IX)}{\det {\mathsf {f}}}}=[{\underline {\mathsf {f}}}(I)]^{-1}{\underline {\mathsf {f}}}(IX).

The concepts of eigenvalues and eigenvectors are somewhat modified. Let $\lambda$ be a real number and let $B$ be a (nonzero) blade of grade $r$ . We say that $B$ is an eigenblade of the function if

\underline {{\mathsf {f}}}(B)=\lambda B,

and $\lambda$ is its eigenvalue. It may seem strange to consider only real eigenvalues, since in linear algebra the eigenvalues of a matrix with all real entries can have complex eigenvalues. In geometric algebra, however, the blades of different orders inherit the complex structure. Since both vectors and pseudovectors can act as eigenblades, they may each have a set of eigenvalues matching the degrees of freedom of the complex eigenvalues that would be found in ordinary linear algebra.

Examples

Simple maps

The identity map and the scalar projection operator are outermorphisms.

Versors

A rotation of a vector by rotor $R$ is given by

f(x)=RxR^{\dagger }

with outermorphism

\underline {{\mathsf {f}}}(X)=RXR^{{\dagger }}.

We check that this is the correct form of the outermorphism. Since rotations are built from the geometric product, which has the distributive property, they must be linear. To see that rotations are also outermorphisms, we recall that rotations preserve angles between vectors:^[5]

x\cdot y=(RxR^{{\dagger }})\cdot (RyR^{{\dagger }})

Next, we try inputting a higher grade element and check that it is consistent with the original rotation for vectors:

{\begin{aligned}\underline {{\mathsf {f}}}(x\wedge y)&=R(x\wedge y)R^{{\dagger }}\\&=R(xy-x\cdot y)R^{\dagger }\\&=RxyR^{\dagger }-R(x\cdot y)R^{{\dagger }}\\&=RxR^{\dagger }RyR^{{\dagger }}-x\cdot y\\&=(RxR^{\dagger })\wedge (RyR^{{\dagger }})+{\cancel {(RxR^{{\dagger }})\cdot (RyR^{{\dagger }})}}-{\cancel {x\cdot y}}\\&=f(x)\wedge f(y)\end{aligned}}

Nonexample – grade projection

An example of a multivector-valued function of multivectors that is linear but is not an outermorphism is grade projection where the grade is nonzero, for example projection onto grade 1:

\langle (x\wedge y)\rangle _{1}=0

\langle x\rangle _{1}\wedge \langle y\rangle _{1}=x\wedge y

References

↑ L. Dorst; C.J.L. Doran; J. Lasenby (2001). Applications of geometric algebra in computer science and engineering. Springer. p. 61. ISBN 0-8176-4267-6.
↑ D. Hestenes; G. Sobczyk (1987). Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. Fundamental Theories of Physics. 5. Springer. p. 68. ISBN 90-277-2561-6.
↑ Except for the case where $f$ is the zero map, when it is required by axiom.
↑ D. Hestenes; G. Sobczyk (1987). Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. Fundamental Theories of Physics. 5. Springer. p. 70. ISBN 90-277-2561-6.
↑ C. Perwass (2008). Geometric Algebra with Applications in Engineering. Geometry and Computing. 4. Springer. p. 23. ISBN 3-540-89067-X.

W.E. Baylis (1996). Clifford (Geometric) Algebras: With Applications in Physics, Mathematics, and Engineering. Springer. p. 71. ISBN 0-8176-3868-7.
A. Crumeyrolle; R. Ablamowicz; P. Lounesto (1995). Clifford Algebras and Spinor Structures: A Special Volume Dedicated to the Memory of Albert Crumeyrolle (1919--1992). Mathematics and Its Applications. 321. Springer. p. 105. ISBN 0-7923-3366-7.
C. D'Orangeville; A. Anthony; N. Lasenby (2003). Geometric Algebra For Physicists. Cambridge University Press. p. 343. ISBN 0-521-48022-1.
P. Joot. Exploring physics with Geometric Algebra. p. 157.

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