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# hw1 - (b Let Î max M denote the largest eigenvalue of the...

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J.M. Gon¸ calves Fall 2003 CDS 212 - Introduction to Modern Control Homework # 1 Date Given: October 2, 2003 Date Due: October 9, 2003 P1. Show that if x R n , the quantity defined by: k x k p := n X k =1 | x k | p ! 1 p is a norm, for 1 p < (Hint: try using H¨ older’s inequality). What happens when p < 1? P2. A set S R n is convex if x, y S λx + (1 - λ ) y S, for all λ [0 , 1] . Show that if k·k is a norm then the unit ball { x R n : k x k ≤ 1 } is convex. P3. Let A C n × n . The matrix norm induced by a vector p -norm is defined as: k A k p := sup x C n ,x 6 =0 k Ax k p k x k p . (a) Show that the expression above is actually a norm on C n × n , by veri- fying the defining properties.
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Unformatted text preview: (b) Let Î» max ( M ) denote the largest eigenvalue of the hermitian matrix M . Show that for the case p = 2, k A k 2 = p Î» max ( A * A ). P4. DFT: Chapter 2, Exercise 1. P5. Prove that the relation h f, g i 2 = Z + âˆž-âˆž g ( t ) * f ( t ) dt satisÂ±es all axioms of the inner product. P6. DFT: Chapter 2, Exercise 2 ( u 2 and u 9 only). P7. DFT: Chapter 2, Exercise 9. P8. DFT: Chapter 2, Exercise 13. P9. Prove entries (1,2), (2,1) and (2,2) in Table 2.2 of DFT....
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