Reduction in the Number of Dimensions

The next question arises immediately how do we determine the coordinates b of the spectral vectors y , in this new system of axes V  

Due to the orthonormality of V, this is a particularly simple linear regression calculation. The vector b is computed as  

Equation (5.15) holds for one specific vector y , . Naturally, it can be expanded into a matrix equation for all y, s in Y. 

All this is not completely new. In Reduced Eigenvector Space (p,180, we did just that the matrix US was used to represent the complete matrix Y. The matrix US we called Yred. The component spectra A can also be represented in the eigenvector axes Ated=AVt. As mentioned then, the reduction in the size of the matrices Y and A can be substantial. 

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