Eigenset generalizations of the eigenvalue concept

For an n-by-n complex matrix A some generalizations of the eigenvalue-eigenvector equation A x = λ x ,     0 ≠ x ∈ C nare investigated. These take the form A S = λ S  or  A S ⊆ λ S where S is a suhset of C n about which various assumptions are made. For example, it is shown that there exists a finite set S ⊂ C n , the sum of whose elements is not 0, such that AS = λS , if and only if λ is an eigenvalue of A in the usual sense. The requirement that the sum of the elements of S is not 0 should he viewed as a natural analog of the requirement x ≠ 0 in the classical eigenvalue-eigenvector equation.