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hw5 - Math508 Fall 2010 Jerry L Kazdan Problem Set 5 D UE...

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Math508, Fall 2010 Jerry L. Kazdan Problem Set 5 D UE : Thurs. Oct. 21, 2010. Late papers will be accepted until 1:00 PM Friday . 1. [Ratio Test] Let a k be a sequence of complex numbers. Ley s : = limsup fl fl fl a k + 1 a k fl fl fl . By comparison with a geometric series, show that the series a k converges absolutely if s < 1. 2. Let A be a square matrix. a) Show that e ( s + t ) A = e sA e tA for all real or complex s , t . b) If AB = BA , the Hoffman text (p. 48) shows that e A + B = e A e B . Give an example showing this may be false if A and B don’t commute. c) If A is any square matrix, show that e A is invertable. d) If A is a 3 × 3 diagonal matrix, compute e A . e) If A and B are similar matrices (so A = S - 1 BS for some invertible matrix S ), show that e A = S - 1 e B S . [In particular, if A is similar to a diagonal matrix D , then by the previous part, e A = S - 1 e D S is easy to compute.] f) If A 2 = 0, compute e A . g) Compute e A for the matrix A = 0 1 2 3 0 0 - 1 - 1 0 0 0 4 0 0 0 0 .
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