Grassmann Algebra
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Grassmann Algebra

Grassmann Algebra

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Explor<strong>Grassmann</strong>Matrix<strong>Algebra</strong>.nb 34<br />

s2A � c2A �� Simplify �� MatrixForm<br />

1 0<br />

�<br />

0 1 �<br />

� Symbolic matrices<br />

B � ��a � x, x � z�, �b, c � z��; MatrixForm�B�<br />

a � x x�z �<br />

b c�z �<br />

<strong>Grassmann</strong>MatrixFunction�Exp�B��<br />

��� a �1 � x� � b ���a � a �a � c �a ��c� x � z<br />

�������������������������������� ���������������� ���������������� �����<br />

�a � c� 2<br />

, ��a ��c� x � z<br />

���������������� ������������������ �,<br />

a � c<br />

1<br />

� ��������������������<br />

�a � c� 3 �b ��a � c���c �a � c � c �� a x � c � a x �<br />

� c x �� a z �� c z � c � c z � a �� a �� c �� a x �� c z�� �<br />

�1 � b� ��2 � a � c � a � 2 � c � c � c � a �� a �� c �� x � z��,<br />

�� c ��1 � z� � b ��a ��c � a �c � c �c � x � z<br />

�������������������������������� ���������������� ������������������<br />

�a � c� 2<br />

��<br />

� Symbolic functions<br />

Suppose we wish to compute the symbolic function f of the matrix A.<br />

A �� MatrixForm<br />

1 � x x�z �<br />

2 3�z �<br />

If we try to compute this we get an incorrect result because the derivatives evaluated at<br />

various scalar values are not recognized by <strong>Grassmann</strong>Simplify as scalars. Here we print<br />

out just three lines of the result.<br />

2001 4 26

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