1. The definition of et is etA=I+tA+ and (a) Use the definition to calculate et, where A: = (b) Also, calculate et for J

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1 The Definition Of Et Is Eta I Ta And A Use The Definition To Calculate Et Where A B Also Calculate Et For J 1 (182.1 KiB) Viewed 13 times

1. The definition of et is etA=I+tA+ and (a) Use the definition to calculate et, where A: = (b) Also, calculate et for J = find all the powers of J. (c) It's a fact that if AB = t² A² 2! (3¹) Now recall that sin(t) = t compute exp 1 (t (²_-2'¹')). (²₂2) + (1 3²), (2) Hint: calculate J², J³, J4 in order to 4³ t5 3! t² cos(t) = 1 2! BA, then eA+B + = 14 4! ∞ n=0 What about exp tn An n! eeB. Use this fact and (a) (b) to ? Hint: (²2¹) († (1 ¹)) ² (1) = (69) for the second one, notice that + (d) Prove that for any invertible matrix M and any matrix A, exp (M(At)M−¹) = M(exp At) M-¹ using the definition.