First Order Asymptotic Expansions for Eigenvalues of Multiplicatively Perturbed Matrices
Fredy Sosa, Julio Moro · SIAM Journal on Matrix Analysis and Applications · 2016
Given an arbitrary square matrix $A$, we obtain the leading terms of the asymptotic expansions in the small, real parameter $\varepsilon$ of multiplicative perturbations $\widehat{A}(\varepsilon)=(I+\varepsilon\,B)A(I+\varepsilon\,C)$ of $A$ for arbitrary matrices $B$ and $C$. The analysis is separated into two rather different cases, depending on whether the unperturbed eigenvalue is zero or not. It is shown that in either case the leading exponents are obtained from the partial multiplicities of the eigenvalue of interest, and the leading coefficients generically involve only appropriately normalized left and right eigenvectors of $A$ associated with that eigenvalue, with no need of generalized eigenvectors. Similar results are obtained for multiplicative perturbation of singular values as well.