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HW1sol - A = 0 So we need to show that A = 0 We are given...

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P ROBLEM SET 1 Partial solution Remark: Note that solutions of odd numbered problems are given at the back of the book. 5. See problem 21 Sec 1.2 6. Let us first consider the case C = 0 . Since the inequality 0 B - 2 Re ( λA ) + | λ | 2 C holds for all lambda it holds for λ = A/C . Therefore 0 B - 2 Re | A | 2 C + | A | 2 C 2 C Multiplying by C yields (since C 0 ) 0 BC - | A | 2 which implies that | A | 2 BC . Now consider the case C = 0 . We need to prove that | A | 2 0 , which holds only when
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Unformatted text preview: A = 0 . So we need to show that A = 0 . We are given that for all λ ∈ C ≤ B-2 Re ( λA ) We prove that A = 0 by contradiction. Assume that A 6 = 0 . Taking λ = B/ A yields that ≤ B-2 Re ± B A A ² =-B which contradicts the assumption that B ≥ . Therefore A = 0 . which completes the proof. 7. See problem 21 Sec 1.1...
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