San José State University Department of Economics 

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Esystems can be vastly generalized beyond the case used here but let us start with indices i,j and k ranging from 1 to 3. First the notion of a permuation of the indices must be considered and the evenness or oddness defined. A permutation of 123 is any rearrangement of the numbers such as 213 or 321. An odd permutation is one that can be achieved with an odd number of transpositions. For example, 213 can be achieved from 123 by transposing the 1 and 2. If the 1 and 3 in 213 are transposed then the result is 231 and thus 231 is an even permutation of 123.
Now the 3x3x3 array e_{ijk} can be defined as
A primary use of esystems is for operations concerning determinants.
In conventional notation the determinant of matrix A whose elements are { a_{ij}} is expressed as
where Cof(a_{ij}), the cofactor of a_{ij}, is obtained by evaluating the determinant of the submatrix formed by deleting the ith row and jth column of the matrix A and multiplying by (−1)^{i+j}. For a 3x3 matrix A the cofactor of a_{11} is (−1)^{2}[a_{22}a_{33}−a_{12}a_{21}. For higher order matrices the formula for the cofactor of an element is impossibly complex.
The determinant in term of esystems is a marvel of simplicity. It is
Thus mathematical manipulations, such as taking the derivative of a determinant, are easy to express.
The generalization of esystems to the case in which the indices range from 1 to n is obvious.
(To be continued.)
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