Selection indices (SI), 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 the others unchanged. Recently a SI was proposed based on the eigenanalysis 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. In this study we developed: (1) a restrictive eigenvalue SI method, RESIM1, that generalizes the ESIM and assigns weights based on the phenotypic variance of the traits; and (2) RESIM2, a generalization of the traditional RSI that assigns weights based on genotypic covariances among traits. We use three data sets to illustrate the theoretical results and practical use of RESIM and its comparison with standard unrestrictive and restrictive selection indices. The main advantages of RESIM1 and RESIM2 over traditional unrestrictive and restrictive SIs are: (1) their statistical sampling properties are known; (2) their responses to selection are equal to or greater than those estimated from the traditional restrictive SI; (3) one traditional restrictive SI is a particular case of RESIM2; and (4) they do not required 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 weights are not available.