Explaining the Generalized Cross-Validation on Linear Models

Cross-Validation is a model validation method widely used by the scientific community. The Generalized Cross-Validation (GCV) is an invariant version of the usual Cross-Validation method. This generalization was obtained using the non usual theory of circulant complex matrices. In this work we intend to give a clear and complete exposition concerning the linear algebra assumptions required by the theory. The aim was to make this text accessible to a wide audience of statisticians and non-statisticians who use the Cross-Validation method in their research activities. It is also intended to supply the absence of a basic reference on this topic in the literature.