STANDARD ERRORS IN THE EIGENVALUES OF A CROSS-PRODUCT MATRIX - THEORYAND APPLICATIONS

New expressions are derived for the standard errors in the eigenvalues of a cross-product matrix by the method of error propagation. Cross-p roduct matrices frequently arise in multivariate data analysis, especi ally in principal component analysis (PCA). The derived standard error s account for the variability in the data as a result of measurement n oise and are therefore essentially different from the standard errors developed in multivariate statistics. Those standard errors were deriv ed in order to account for the finite number of observations on a fixe d number of variables, the so-called sampling error. They can be used for making inferences about the population eigenvalues. Making inferen ces about the population eigenvalues is often not the purposes of PCA in physical sciences. This is particularly true if the measurements ar e performed on an analytical instrument that produces two-dimensional arrays for one chemical sample: the rows and columns of such a data ma trix cannot be identified with observations on variables at all. Howev er, PCA can still be used as a general data reduction technique, but n ow the effect of measurement noise on the standard errors in the eigen values has to be considered. The consequences for significance testing of the eigenvalues as well as the usefulness for error estimates for scores and loadings of PCA, multiple linear regression (MLR) and the g eneralized rank annihilation method (GRAM) are discussed. The adequacy of the derived expressions is tested by Monte Carlo simulations.