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Question: one of the best known results of eigenvalue perturbation theory...

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One of the best known results of eigenvalue perturbation theory is the Bauer-Fike theorem. Suppose A E C^mxm is diagonalizable with A = V AV^-1, and let

\delta A\in C^{m*m}

be arbitrary. Then every eigenvalue of A+\delta A lies in at least one of the m circular disks in the complex plane of radius \kappa (V)\left \| \delta A \right \|_{2} centered at the eigenvalues of A, where \kappa is the 2-norm condition number

Suppose A is normal. Show that for each eigenvalue \bar{\lambda _{j}} of A+\delta A , there is an eigenvalue \lambda _{j} , of A such that

\left | \bar{\lambda _{j}}-\lambda _{j} \right | \leq \left \| \delta A \right \|_{2}

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