Sensitivity analysis of all eigenvalues of a symmetric matrix
Let \(A = A(x)\) be a symmetric \(n \times n\) matrix whose elements depend smoothly on the \(p\)-vector \(x\). This paper studies the sensitivity of the eigenvalues \(\lambda_ k (x) = \lambda_ k (A(x))\) as a function of \(x\). Because \(\lambda_ k (x)\) is not convex, the functions \(\sigma_ m(x) = \sum_{k=1}^ m \lambda_ k (x)\) are introduced which turn out to be convex. Since moreover \(\lambda_ k (x)\) and \(\sigma_ m(x)\) are nonsmooth, the derivations depend heavily on results from nonsmooth and convex analysis. Another important element in the analysis is the separation of the effects of the maps \(A \mapsto \lambda_ m(A)\) and of \(x \mapsto A(x)\). First \(\sigma_ m\) is characterized by Ky Fan's variational principle, i.e., \(\sigma_ m = \max \{\text{tr} (A XX^ T)\}\) where the maximum is taken over all \(n \times m\) matrices \(X\) satisfying \(X^ TX = I_ m\). Next the subdifferential calculus for the map \(A \mapsto \lambda_ m (A)\) is analysed. Finally the generalized gradient and the directional derivatives for \(\lambda_ m(x)\) are explicitly computed.
scientific article; zbMATH DE number 739396