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

edit

Let   be an  -linear map from   to  . The extension of   to an outermorphism is the unique map   satisfying

 
 
 
 

for all vectors   and all multivectors   and  , where   denotes the exterior algebra over  . 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  ,   and vectors  ,  ,  , the outermorphism is linear over bivectors:

 

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

Adjoint

edit

Let   be an outermorphism. We define the adjoint of   to be the outermorphism that satisfies the property

 

for all vectors   and  , where   is the nondegenerate symmetric bilinear form (scalar product of vectors).

This results in the property that

 

for all multivectors   and  , where   is the scalar product of multivectors.

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

 

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

edit

It follows from the definition at the beginning that the outermorphism of a multivector   is grade-preserving:[2]

 

where the notation   indicates the  -vector part of  .

Since any vector   may be written as  , it follows that scalars are unaffected with  .[b] Similarly, since there is only one pseudoscalar up to a scalar multiplier, we must have  . The determinant is defined to be the proportionality factor:[3]

 

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:

 

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

 

and so does its adjoint, with

 

The concepts of eigenvalues and eigenvectors may be generalized to outermorphisms. Let   be a real number and let   be a (nonzero) blade of grade  . We say that a   is an eigenblade of the function with eigenvalue   if[4]

 

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 grades can exhibit a 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

edit
Simple maps

The identity map and the scalar projection operator are outermorphisms.

Versors

A rotation of a vector by a rotor   is given by

 

with outermorphism

 

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]

 

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

 
Orthogonal projection operators

The orthogonal projection operator   onto a blade   is an outermorphism:

 
Nonexample – orthogonal rejection operator

In contrast to the orthogonal projection operator, the orthogonal rejection   by a blade   is linear but is not an outermorphism:

 
Nonexample – grade projection operator

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:

 
 

Notes

edit
  1. ^ See particularly Exterior algebra § Functoriality.
  2. ^ Except for the case where   is the zero map, when it is required by axiom.

Citations

edit

References

edit
  • Hestenes, D.; Sobczyk, G. (1987), Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, Fundamental Theories of Physics, vol. 5, Springer, ISBN 90-277-2561-6
  • Crumeyrolle, A.; Ablamowicz, R.; Lounesto, P. (1995), Clifford Algebras and Spinor Structures: A Special Volume Dedicated to the Memory of Albert Crumeyrolle (1919–1992), Mathematics and Its Applications, vol. 321, Springer, p. 105, ISBN 0-7923-3366-7
  • Baylis, W.E. (1996), Clifford (Geometric) Algebras: With Applications in Physics, Mathematics, and Engineering, Springer, p. 71, ISBN 0-8176-3868-7
  • Dorst, L.; Doran, C.J.L.; Lasenby, J. (2001), Applications of geometric algebra in computer science and engineering, Springer, p. 61, ISBN 0-8176-4267-6
  • D'Orangeville, C.; Anthony, A.; Lasenby, N. (2003), Geometric Algebra For Physicists, Cambridge University Press, p. 343, ISBN 0-521-48022-1
  • Perwass, C. (2008), Geometric Algebra with Applications in Engineering, Geometry and Computing, vol. 4, Springer, p. 23, ISBN 3-540-89067-X
  • Joot, P. (2014), Exploring physics with Geometric Algebra, p. 157