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26.3. One of the best known results of eigenvalue perturbation theory is the Bauer-Fike theorem. Suppose A € Cmxm is diagonalizable with A = VAV-¹, and let SA € Cmxm be arbitrary. Then every eigenvalue of A + SA lies in at least one of the m circular disks in the complex plane of radius <centered at the eigenvalues of A, where K is the 2-norm condition number. (Compare Exercise 24.2.) (a) Prove the Bauer-Fike theorem by using the equivalence of conditions (i) and (iv) of Exercise 26.1. (b) Suppose A is normal. Show that for each eigenvalue i, of A + 6A, there is an eigenvalue Ij of A such that 11; - Ail - < ||6A||2- (26.4)

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