Outermorphism
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. Definition. Let formula_1 be an formula_2-linear map from formula_3 to formula_4. The extension of formula_1 to an outermorphism is the unique map formula_6 satisfying for all vectors formula_11 and all multivectors formula_12 and formula_13, where formula_14 denotes the exterior algebra over formula_3. 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 formula_16, formula_17 and vectors formula_11, formula_19, formula_20, the outermorphism is linear over bivectors: which extends through the axiom of distributivity over addition above to linearity over all multivectors. Adjoint. Let formula_22 be an outermorphism. We define the "adjoint" of formula_23 to be the outermorphism that satisfies the property for all vectors formula_25 and formula_26, where formula_27 is the nondegenerate symmetric bilinear form (scalar product of vectors). This results in the property that for all multivectors formula_12 and formula_13, where formula_31 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. It follows from the definition at the beginning that the outermorphism of a multivector formula_12 is grade-preserving: where the notation formula_35 indicates the formula_36-vector part of formula_12. Since any vector formula_11 may be written as formula_39, it follows that scalars are unaffected with formula_40. Similarly, since there is only one pseudoscalar up to a scalar  multiplier, we must have formula_41. The determinant is defined to be the proportionality factor: 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 formula_46 be a "real" number and let formula_13 be a (nonzero) blade of grade formula_36. We  say that a formula_13 is an eigenblade of the function with eigenvalue formula_46 if 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. The identity map and the scalar projection operator are outermorphisms. A  rotation of a vector by a rotor formula_52 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: Next, we try inputting a higher grade element and check that it is consistent with the original rotation for vectors: The orthogonal projection operator formula_57 onto a blade formula_13 is an outermorphism: In contrast to the orthogonal projection operator, the orthogonal rejection formula_60 by  a blade formula_13 is linear but is "not" an outermorphism: 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: