On the sensitivity analysis of eigenvalues

Let λ be a simple eigenvalue of an n-by-n matrix A. Let y and x be left and right eigenvectors of A corresponding to λ, respectively. Then, for the spectral norm, the condition number cond(λ,A) := ‖x‖2 ‖y‖2/|yx| measures the sensitivity of λ to small perturbations in A and plays an important role in the accuracy assessment of computed eigenvalues. R. A. Smith [Numer. Math., 10(1967), pp.232-240] proved that cond(λ,A) = ‖x‖2‖y‖2/|yx| = ‖adj(λI − A)‖2/|p(λ)|, where adj(A) is the “adjugate” of A and p(λ) is the derivative of p(z) := det(zI − A) at λ. We extend Smith’s condition number to any matrix norm ‖ · ‖ and show that cond(λ,A) = ‖yx‖∗ |y∗x| = ‖adj(λI −A)‖∗ |p′(λ)| measures the sensitivity of λ to small perturbations in A, where ‖ · ‖∗ is the dual norm of ‖ · ‖. The matlab command roots computes roots of a polynomial p(x) by computing the eigenvalues of a companion matrix Cp associated with p. We analyze the sensitivity of λ as a root of p(x) as well as the sensitivity of λ as an eigenvalue of Cp and compare their condition numbers.