Let A be a matrix with eigenvalues 1, . . . , n and let be an eigenvalue of A + E. Let X

Chapter 7, Problem 9

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Let A be a matrix with eigenvalues 1, . . . , n and let be an eigenvalue of A + E. Let X be a matrix that diagonalizes A and let C = X1EX. Prove: (a) For some i , | i| _n j=1 |ci j | [Hint: is an eigenvalue of X1(A+E)X. Apply Gerschgorins theorem from Exercise 7.] (b) min 1jn | j| cond(X)_E_ 1

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