We construct a real symmetric matrix with known eigenvalues by using the QR factorization to produce a random orthogonal set of eigenvectors.
n = 30;
lambda = (1:n)';
D = diag(lambda);
[V,R] = qr(randn(n)); % get a random orthogonal V
A = V*D*V';
The condition number of these eigenvalues is one. Thus the effect on them is bounded by the norm of the perturbation to 
.
. for k = 1:3
E = randn(n); E = 1e-4*E/norm(E);
mu = sort(eig(A+E));
max_change = norm(mu-lambda,inf)
end