  The eigenvalue problem for a linear function L centers on solving the
eigen-equation Lx = rx. This paper generalizes the eigenvalue problem from a
single linear function to an iterated function system F consisting of possibly
an infinite number of linear or affine functions. The eigen-equation becomes
F(X) = rX, where r>0 is real, X is a compact set, and F(X)is the union of f(X),
for f in F. The main result is that an irreducible, linear iterated function
system F has a unique eigenvalue r equal to the joint spectral radius of the
functions in F and a corresponding eigenset S that is centrally symmetric,
star-shaped, and full dimensional. Results of Barabanov and of
Dranishnikov-Konyagin-Protasov on the joint spectral radius follow as
corollaries.
