if T^{2}= T, then T satisfies the equation x^{2}- x = 0, which factors as x(x - 1) = 0.

thus the minimal polynomial for T divides x^{2}- x, so is either:

a) x (in which case T is the 0-matrix), and thus T has only 0 eigenvalues.

b) x-1 (in which case T is the identity matrix), and thus T has only 1 eigenvalues,

c) x(x-1), in which case T has both 0 and 1 for eigenvalues.

the eigenspace corresponding to the eigenvalue 0 is called the null space (or kernel) of T. (for what is an eigenvector in this space? it is a non-zero vector v such that T(v) = 0v = 0).

the eigenspace corresponding to the eigenvalue 1 must (in this case) be the range of T (for if for a non-zero w, we have w = T(v), then T(w) = T(T(v)) = T^{2}(v) = T(v) = w, so w is an eigenvector with eigenvalue 1).