Proposition 6. Assume that V is finite-dimensional. Let TE L(V) and 1, ..., Am EF be a list of distinct eigenvalues. Then the following are equivalent: (1) there exist a basis By such that M(Bv, T) is a diagonal matrix; (2) there exist a basis By consisting of eigenvectors of T; (3) V = E(1, T) O . .. DE( m, T); (4) dim V = dim E(M1, T) + .. . + dim E(m, T). Proof. Write this! 0