Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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 $\frac{\partial}{\partial\varepsilon}\det(\lambda I-A(\varepsilon))|_{(\varepsilon,\lambda)=(0,\lambda_{0})}\neq0$,  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 $\lambda_{0}$ and their corresponding eigenvectors can be represented by a single convergent Puiseux series containing only pors of $\varepsilon^{1/m}$, and there are explicit recursive formulas to compute all the Puiseux series coefficients from just the derivatives of $A(\varepsilon)$ at the origin. Using these recursive formulas  calculate the series coefficients up to the second order and list them for quick reference.  gives, under a generic condition, explicit recursive formulas to compute the perturbed eigenvalues and eigenvectors for non-self-adjoint analytic perturbations of matrices with nonderogatory eigenvalues.
['Aaron Welters']
I am writing a research paper on a idea On Explicit Recursive Formulas in the Spectral Perturbation Analysis of a Jordan Block
[{"from": "human", "value": "I am writing a research paper on a idea On Explicit Recursive Formulas in the Spectral Perturbation Analysis of a Jordan Block"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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 $\\frac{\\partial}{\\partial\\varepsilon}\\det(\\lambda I-A(\\varepsilon))|_{(\\varepsilon,\\lambda)=(0,\\lambda_{0})}\\neq0$,  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 $\\lambda_{0}$ and their corresponding eigenvectors can be represented by a single convergent Puiseux series containing only pors of $\\varepsilon^{1/m}$, and there are explicit recursive formulas to compute all the Puiseux series coefficients from just the derivatives of $A(\\varepsilon)$ at the origin. Using these recursive formulas  calculate the series coefficients up to the second order and list them for quick reference.  gives, under a generic condition, explicit recursive formulas to compute the perturbed eigenvalues and eigenvectors for non-self-adjoint analytic perturbations of matrices with nonderogatory eigenvalues."}]