Algebra of multi-way analysis

The algebra of multi-way arrays is described in a field of mathematics called tensor analysis, which is an extension and generalization of matrix algebra. A zero-order tensor is a scalar a first-order tensor is a vector a second-order tensor is a matrix a third-order tensor is a three-way array a fourth-order tensor is a four-way array and so on. The notions of addition, subtraction and multiplication of matrices can be generalized to multi-way arrays. This is shown in the following sections [Borisenko Tarapov 1968, Budiansky 1974], 

Addition and subtraction of multi-way arrays is a direct generalization of that of two-way arrays. Suppose that X and Y have the same dimensions (/ x J x K) then Z = X + Y is defined by 

Addition and subtraction are only defined for arrays of the same orders and dimensions. Generalizations to sums of more than two multi-way arrays are straightforward. 

It is also possible to have an outer product of more than two arrays. Consider taking the outer product of x (7 x 1), y (J x 1) and z (K x 1). Then using the definition in Equation 

The same example as used before in Example 2.5 is considered. 

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