For example, in Matlab we can define a linear function "fun" to map the m^2 dimensional vector X to another m^2 dimensional vector as follows:
m=3;
A=rand(m,m);
fun = @(X)reshape(reshape(X,[m,m])*A,[m^2,1]);
T = eigs(fun,m^2,1)
Then the function "eigs" can solve the eigenvalue T of this function without specifying the exact matrix form of '"fun".
This mapping can involve more tensors making the construction of the whole matrix expensive, while in Mathematica, it seems "Eigenvalues" can not deal with this kind of function.