We present an algorithm for the reduction of dimensionality useful in statistical classification problems where observations from two multivariate normal distributions are discriminated. It is based on Principal Components Analysis and consists of a simultaneous diagonalization of two covariance matrices. The criterion for reduction of dimensionality is given by the contribution of each principal component to the area under the ROC curve of a discriminant function. Linear and quadratic scores are considered, the focus being on the quadratic case.