Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 3
tedious to define all the scalar factors involved. We then discuss the common algebraic
operations like multiplication by scalars, addition, multiplication, taking the complement,
finding the grade of elements, simplification, taking components or determining the type of
elements involved.
Separate sections are devoted to discussing the notions of transpose, determinant and adjoint
of a Grassmann matrix. These notions do not carry over directly into the algebra of
Grassmann matrices due to the non-commutative nature of Grassmann numbers.
Matrix powers are then discussed, and the inverse of a Grassmann matrix defined in terms of
its positive integer powers. Non-integer powers are defined for matrices whose bodies have
distinct eigenvalues. The determination of the eigensystem of a Grassmann matrix is
discussed for this class of matrix.
Finally, functions of Grassmann matrices are discussed based on the determination of their
eigensystems. We show that relationships that we expect of scalars, for example
Log[Exp[A]] = A or Sin�A� 2 � Cos�A� 2 � 1, can still apply to Grassmann matrices.
13.2 Creating Matrices
� Creating general symbolic matrices
Unless a Grassmann matrix is particularly simple, and particularly when a large symbolic
matrix is required, it is useful to be able to generate the initial form of a the matrix
automatically. We do this with the GrassmannAlgebra CreateMatrixForm function.
? CreateMatrixForm
CreateMatrixForm�D��X,S� constructs an array of the specified
dimensions with copies of the expression X formed by indexing
its scalars and variables with subscripts. S is an optional
list of excluded symbols. D is a list of dimensions of
the array �which may be symbolic for lists and matrices�.
Note that an indexed scalar is not recognised as a scalar
unless it has an underscripted 0, or is declared as a scalar.
Suppose we require a 2�2 matrix of general Grassmann numbers of the form X (given
below) in a 2-space.
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