Product notation

Complex scalar products arise naturally in quantum mechanics because there is an experimental interpretation for the complex scalar product of two wave functions (as we saw in Section 1.2). Students of physics should note that the traditional brac-ket notation is consistent with our complex scalar product notation—just put a bar in place of the comma. The physical importance of the bracket will allow us to apply our intuition about Euclidean geometry (such as orthogonality) to states of quantum systems. 

Equation (41) can be viewed as the reciprocal of Eq. (37). It can be rewritten, using convolution product notations, as... 

The short-hand double dot product notation resembles (1.118) and is de-flned by ... 

For the previous small numerical example, the outer product notation of the PCA model of X becomes... 

In Kronecker product notation the Tucker3 model reads... 

Using the tensor products notation (see Chapter 2) an alternative expression of the PARAFAC model is ... 

Using the tensor product notation it becomes clear that the Tucker3 model generalizes the PARAFAC model. Not only tensor products of the same components in all three modes are taken, but also their cross-products. Moreover, the Tucker3 model allows for a different number of components in all three modes. 

The Tuckerl model in Kronecker products notation reads ... 

All production notations must be legibly entered and readily traceable. Batch records shall be retained for the same period as the raw materials and finished product samples. 

Note that these equations can also be written without resuming to the jU-product notation indicated by the round brackets. Instead one can use common Dirac notation of the inner product in the Hilbert space Y which is indicated by the use of angular brackets like usual. In this case the metric operator fi (defined in Sec. IIB) appears explicitly and, e.g., Eq. (26) becomes... 

The product notation n is analogous to E for sums. More explicitly. 

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