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-
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