On Explicit Recursive Formulas in the Spectral Perturbation Analysis of a Jordan Block
On Explicit Recursive Formulas in the Spectral Perturbation Analysis of a Jordan Block
Let $A(\varepsilon)$ be an analytic square matrix and $\lambda_{0}$ an eigenvalue of $A(0)$ of algebraic multiplicity $m\geq1$. Then under the condition $\frac{\partial}{\partial\varepsilon}\det(\lambda I-A(\varepsilon))|_{(\varepsilon,\lambda)=(0,\lambda_{0})}\neq0$, we prove that the Jordan normal form of $A(0)$ corresponding to the eigenvalue $\lambda_{0}$ consists of a single $m\times m$ Jordan block, the perturbed eigenvalues near $\lambda_{0}$ …