Let N be a n×n normal matrix with eigenvalues v_1, v_2, …, v_n, and let A be an×n diagonalizable matrix, i. e., there is a nonsingular matrix Q such that Q~(-1)AQ=diag (α_1, α_2, …, α_n). It is proved that there exists a suitable permutation π of the set{1,2, …, n} such that(sum from i=1 to n |v_i-α_(π(i))|~2)~(1/2)≤||Q||_2||Q~(-1)||_2||A-N||_F,where || ||_2 denotes the spectral norm, and || ||_F the Frobenius norm.