Rank and Eigenvalues

Note that there is sometimes a difference in definition in the literature between the exact mathematical rank (which involves finding the number of PCs that precisely model a dataset with no residual error) and the approximate (or chemical) rank which involves defining the number of significant components in a dataset we will be concerned with this later in the chapter. 

Normally after PCA, the size of each component can be measured. This is often called an eigenvalue the earlier (and more significant) the components, the larger their size. There are a number of definitions in the literature, but a simple one defines the eigenvalue of a PC as the sum of squares of die scores, so that 

The sum of all nonzero eigenvalues for a datamatrix equals the sum of squares of the entire data-matrix, so that 

The cumulative percentage eigenvalue is often used to determine (approximately) what proportion of die data has been modelled using PCA and is given by Y. =lga. The closer to 100 %, die more faithful is die model. The percentage can be plotted against the number of eigenvalues in die PC model. 

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Both PCA and Classical MDS give rise to the same low-dimensional embedding and the Gram matrix (Eq. 2.4) has the same rank and eigenvalues up to a constant factor as the feature (covariance) matrix of PCA [5]. 

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