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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 =4fora l l k ≥ 3. (ii) Rank( A - 2 I ) = 7 and rank( A - 2 I ) k =6fora l l k ≥ 2. (iii) Rank( A - 3 I ) k =6fora l l k ≥ 1. Find the Jordan form of A . 2. (a) Prove that any linear transformation T of a ﬁnite-dimensional C -vector space V canbew r i t teninthefo rm 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
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This document was uploaded on 01/25/2012.
- Spring '09