Proof: Let mi be algebraic multiplicity, di be geometric multiplicity, and n = dim V.
Suppose diagonalizable. Then let B be a basis consisting of eigenvectors. Intersect this basis with each eigenspace. Call it Bi. Let ni = Bi. Then n ≤ ∑ni ≤ ∑di ≤ ∑mi ≤ n.
So ∑(mi − di) = 0. But each term must be nonnegative.
Conversely, suppose the multiplicties are equal. Then take union of eigenbasis. They form a basis for full space.