On the sensitivity analysis of eigenvalues

Abstract

Let $ \lam$ be a simple eigenvalue of an $n$-by-$n$ matrix $A.$ Let $y$ and $ x$ be left and right eigenvectors of $A$ corresponding to $\lam,$ respectively. Then, for the spectral norm, the condition number $\cond(\lam, A) := \|x\|_2\, \|y\|_2 /{|y^*x|}$ measures the sensitivity of $\lam$ 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(\lam, A) = \|x\|_2\|y\|_2/{|y^*x|} = \|\adj(\lam I -A)\|_2/{|p'(\lam)|}$, where $ \adj(A)$ is the ``adjugate" of $A$ and $p'(\lam)$ is the derivative of $ p(z) :=\det(z I- A)$ at $\lam.$ We extend Smith's condition number to any matrix norm $\|\cdot\|$ and show that $$\cond(\lam, A) = \frac{\|yx^*\|_*}{|y^*x|} = \frac{\|\adj(\lam I - A)^*\|_*}{|p'(\lam)|}$$ measures the sensitivity of $\lam$ to small perturbations in $A,$ where $\norm_*$ is the dual norm of $\|\cdot\|.$ The {\sc matlab} command {\tt roots} computes roots of a polynomial $p(x)$ by computing the eigenvalues of a companion matrix $C_p$ associated with $p.$ We analyze the sensitivity of $\lam$ as a root of $p(x)$ as well as the sensitivity of $\lam$ as an eigenvalue of $C_p$ and compare their condition numbers.