Triple Products of Vectors

FIGURE 11.6 Inversion behavior of polar vector P and axial vector w. Axes and vectors before and after inversion are drawn. 

FIGURE 11.7 Triple scalar product as volume of parallelepiped base area BCsvaO, altitude = Acos p, volume = ABCsin0cos = A (B X C).  

For three polar vectors, the triple scalar product changes sign upon inversion. Such a quantity is known as a pseudoscalar, in contrast to a scalar, which is invariant to inversion. 

You might also encounter the triple vector product A x (B x C), which is itself a vector quantity. This can be evaluated using the Levi-Civita representation (11.31). The i component of the triple product can be written as 

Since a sum containing a Kronecker delta reduces to a single term, we find 

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In principal component analysis (PCA), a matrix is decomposed as a sum of vector products, as shown in Figure 1.6. The vertical vectors (following the object way) are called scores and the horizontal vectors (following the variable way) are called loadings. A similar decomposition is given for three-way arrays. Here, the array is decomposed as a sum of triple products of vectors as in Figure 1.7. This is the PARAFAC model. The vectors, of which there are three different types, are called loadings. 

Figure 1.7. In PARAFAC, a three-way array is decomposed into a sum of triple products of vectors. The vectors are called loadings. There are three different types called A,B and C for convenience.

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