Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 27
13.9 Matrix Eigensystems
Exterior eigensystems of Grassmann matrices
Grassmann matrices have eigensystems just as do real or complex matrices. Eigensystems
are useful because they allow us to calculate functions of matrices. One major difference in
approach with Grassmann matrices is that we may no longer be able to use the determinant
to obtain the characteristic equation, and hence the eigenvalues. We can, however, return to
the basic definition of an eigenvalue and eigenvector to obtain our results. We treat only
those matrices with distinct eigenvalues. Suppose the matrix A is n�n and has distinct
eigenvalues, then we are looking for an n�n diagonal matrix L (the matrix of eigenvalues)
and an n�n matrix X (the matrix of eigenvectors) such that:
A � X � X � L
13.6
The pair {X,L} is called the eigensystem of A. If X is invertible we can post-multiply by X �1
to get:
and pre-multiply by X �1 to get:
A � X � L � X �1
X �1 � A � X � L
13.7
13.8
It is this decomposition which allows us to define functions of Grassmann matrices. This
will be discussed further in Section 13.10.
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