Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 2
13.6 Matrix Powers
� Positive integer powers
� Negative integer powers
� Non-integer powers of matrices with distinct eigenvalues
� Integer powers of matrices with distinct eigenvalues
13.7 Matrix Inverses
A formula for the matrix inverse
� GrassmannMatrixInverse
13.8 Matrix Equations
� Two types of matrix equations
� Solving matrix equations
13.9 Matrix Eigensystems
Exterior eigensystems of Grassmann matrices
� GrassmannMatrixEigensystem
13.10 Matrix Functions
Distinct eigenvalue matrix functions
� GrassmannMatrixFunction
� Exponentials and Logarithms
� Trigonometric functions
� Symbolic matrices
� Symbolic functions
13.11 Supermatrices
To be completed
13.1 Introduction
This chapter introduces the concept of a matrix of Grassmann numbers, which we will call a
Grassmann matrix. Wherever it makes sense, the operations discussed will work also for
listed collections of the components of tensors of any order as per the Mathematica
representation: a set of lists nested to a certain number of levels. Thus, for example, an
operation may also work for vectors or a list containing only one element.
We begin by discussing some quick methods for generating matrices of Grassmann
numbers, particularly matrices of symbolic Grassmann numbers, where it can become
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