Given a set "S" of matrices, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of "S". Equivalently, for any set "S" of mutually commuting semisimple linear transformations of a finite-dimensional vector space "V" there exists a basis of "V" consisting of "simultaneous eigenvectors" of all elements of "S". Each of these common eigenvectors "v" ∈ "V" defines a linear functional on the subalgebra "U" of End("V") generated by the set of endomorphisms "S"; this functional is defined as the map which associates to each element of "U" its eigenvalue on the eigenvector "v". This map is also multiplicative, and sends the identity to 1; thus it is an algebra homomorphism from "U" to the base field. This "generalized eigenvalue" is a prototype for the notion of a weight.