My professor states that "A matrix is diagonalizable if and only if the sum of the geometric multiplicities of the eigen values equals the size of the matrix". I have to prove this and proofs are my biggest weakness; but, I understand that geometric multiplicites means the dimensions of the solution space for the equation Ax=λx (right?). But what does the "sum of the geometric multiplicities" mean? Could you point me in the right direction, thanks!!