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.
It follows from the definition at the beginning that the outermorphism of a multivector is grade-preserving:
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 pseudoscalarup to a scalar multiplier, we must have . 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 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
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.
The identity map and the scalar projection operator are outermorphisms.
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:
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:
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