In geometric algebra, the outermorphism of a linear function between vector spaces is a natural extension of the map to arbitrary multivectors. It is the unique unital algebra homomorphism of exterior algebras whose restriction to the vector spaces is the original function.
Contents
Definition
Let
for all vectors
The outermorphism inherits linearity properties of the original linear map. For example, we see that for scalars
which extends through the axiom of distributivity over addition above to linearity over all multivectors.
Adjoint
Let
for all vectors
This results in the property that
for all 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
It follows from the definition at the beginning that the outermorphism of a multivector
where the notation
Since any vector
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 are somewhat modified. Let
and
Examples
The identity map and the scalar projection operator are outermorphisms.
A rotation of a vector by rotor
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:
Next, we try inputting a higher grade element and check that it is consistent with the original rotation for vectors:
The orthogonal projection operator
In contrast to the orthogonal projection operator, the orthogonal rejection
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: