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MA554_AUG97 - Math 554 Qualifying Examination August 1997...

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Math 554 Qualifying Examination August, 1997 In answering any part of a question you may assume the preceding parts. 1. Let A be an 8 × 8 complex matrix satisfying the following conditions (with I an 8 × 8 identity matrix): (i) Rank( A + I ) = 6, rank( A + I ) 2 = 5, and rank( A + I ) k = 4 for all k 3. (ii) Rank( A - 2 I ) = 7 and rank( A - 2 I ) k = 6 for all k 2. (iii) Rank( A - 3 I ) k = 6 for all k 1. Find the Jordan form of A . 2. (a) Prove that any linear transformation T of a finite-dimensional C -vector space V can be written in the form T = S + N where S is diagonalizable, N is nilpotent, and SN = NS . (Hint: Look at the Jordan form.) (b) Use (without proof) the equivalence of “ S diagonalizable” and “the minimal polynomial of S has no multiple factors” to show that if S is diagonalizable and if W is an S -invariant subspace of V then the restriction of S to W is diagonalizable. (c) Let T = S + N be as in (a). Let I be the identity transformation of V . For any λ C , and any m > 0, let W λ,m be the kernel of ( T - λI ) m . Show that the restriction of ( S - λI
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