Grassmann Algebra

TheExteriorProduct.nb 24
� 1 The determinant changes sign on the interchange of any two rows.
The exterior product is anti-symmetric in any two factors.
� � Αi � � � Αj � � ��� � Αj � � � Αi � �
� 2 The determinant is zero if two of its rows are equal.
The exterior product is nilpotent.
� � Αi � � � Αi � � � 0
� 3 The determinant is multiplied by a factor k if any row is multiplied by k.
The exterior product is commutative with respect to scalars.
Α1 � � ��k Αi��� � k�� Α1 � � � Αi � ��
� 4 The determinant is equal to the sum of p determinants if each element of a given row
is the sum of p terms.
The exterior product is distributive with respect to addition.
Α1 � � ��� i Αi��� � � i �� Α1 � � � Αi � ��
� 5 The determinant is unchanged if to any row is added scalar multiples of other rows.
The exterior product is unchanged if to any factor is added multiples of other factors.
Α1 � � ��Αi � �
j�i �k j �Αj��� � Α1 � � � Αi � �
The Laplace expansion technique
The Laplace expansion technique is equivalent to the calculation of the exterior product in four
stages:
1. Take the exterior product of any p of the Αi .
2. Take the exterior product of the remaining n-p of the Αi .
3. Take the exterior product of the results of the first two operations.
4. Adjust the sign to ensure the parity of the original ordering of the Αi is preserved.
Each of the first two operations produces an element with � n n
� = � � terms.
p n � p
A generalization of the Laplace expansion technique is evident from the fact that the exterior
product of the Αi may be effected in any grouping and sequence which facilitate the
computation.
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