Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 25
Finally, if we try to find the inverse of a matrix with a singular body we get the expected
type of error messages.
A � ��1 � x, 2 � y�, �1 � z, 2 � w��; MatrixForm�A�
1 � x 2�y �
1 � z 2�w �
GrassmannMatrixInverse�A�
Inverse ::sing : Matrix ��1, 2�, �1, 2�� is singular .
13.8 Matrix Equations
� Two types of matrix equations
If we can recast a system of equations into a matrix form which maintains the correct
ordering of the factors, we can use the matrix inverse to obtain the solution. Suppose we
have an equation M�X�Y where M is an invertible matrix of Grassmann numbers, X is a
vector of unknown Grassmann numbers, and Y is a vector of known Grassmann numbers.
Then the solution X is given by:
����GrassmannMatrixInverse�M��Y��
Alternatively, if the equations need to be expressed as X�M�Y in which case the solution X is
given by:
����Y � GrassmannMatrixInverse�M���
GrassmannAlgebra implements these as GrassmannLeftLinearSolve and
GrassmannRightLinearSolve respectively.
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? GrassmannLeftLinearSolve
GrassmannLeftLinearSolve�M,Y� calculates a
matrix or vector X which solves the equation M�X��Y.