On the Correlation between Random Variables and their Principal
  Components
Zenon Gniazdowski
The article attempts to find an algebraic formula describing the correlation
coefficients between random variables and the principal components representing
them. As a result of the analysis, starting from selected statistics relating
to individual random variables, the equivalents of these statistics relating to
a set of random variables were presented in the language of linear algebra,
using the concepts of vector and matrix. This made it possible, in subsequent
steps, to derive the expected formula. The formula found is identical to the
formula used in Factor Analysis to calculate factor loadings. The discussion
showed that it is possible to apply this formula to optimize the number of
principal components in Principal Component Analysis, as well as to optimize
the number of factors in Factor Analysis.