We will confirm the Bauer-Fike theorem on a triangular matrix. These tend to be far from normal.
n = 15;
lambda = (1:n)';
A = triu( ones(n,1)*lambda' );
The Bauer-Fike theorem provides an upper bound on the condition number of these eigenvalues.
[V,D] = eig(A);
kappa = cond(V)
The theorem suggests that eigenvalue changes may be up to 7 orders of magnitude larger than a perturbation to the matrix. A few random experiments show that effects of nearly that size are not hard to observe.
for k = 1:3
E = randn(n); E = 1e-7*E/norm(E);
mu = eig(A+E);
max_change = norm( sort(mu)-lambda, inf )
end