||Selection indices, used in animal and plant breeding to select the best individuals for the next breeding cycle, are based on phenotypic observations of traits recorded in candidate individuals. The restrictive selection index (RSI) facilitates maximizing the genetic progress of some characters, while leaving others unchanged. Recently a selection index (SI) was proposed based on the eigen analysis method (ESIM), in which the first eigenvector (from the largest eigenvalue) is used as the SI criterion, and its elements determine the proportion of the trait that contributes to the SI. However,the current ESIM, which has two main limitations, is based on the assumption that the vector of coefficients of the index is equal to the genotypic variance-covariance matrix among the traits multiplied by the vector of economic weights, and does not allow one to restrict the number of traits. In this study, we develop a more general ESIM that has two main features, namely, it makes no assumption concerning the coeffcients of the index and it can be generalized to a restrictive ESIM (RESIM). We use two datasets to illustrate the theoretical results and practical use of ESIM and RESIM, and to compare them with standard unrestrictive and restrictive selection indices. The main advantages of RESIM over traditional unrestrictive and restrictive SIs are that its statistical sampling properties are known; its selection responses are equal to or greater than those estimated from the traditional restrictive SI; and it does not require economic weights and thus can be used in practical applications when all or some of the traits need to be improved simultaneously (traditional SIs cannot improve several traits simultaneously if economic weights are not available). Finally, we prove that the coefficients of the traditional RSI belong to the space generated by the eigenvectors of RESIM.