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Task 9 (Matrix Exponential I) Let T: (0, oc) → L(C") satisfy • T(0) = Idx; • T(t + s) = T(t)T(s) (t, s 2 0); •T€ C(0, 0), L(C")). Show that T is differentiable (from the right) in 0. Show that T(t) = e' A, where A = †(0). Hint: Show that v(t) := T(s) ds is invertible for small to and for those T(t) = V(to)(V(t +to) - V(t)).

Task 9 (Matrix Exponential I) Let T: (0, oc) → L(C") satisfy • T(0) = Idx; • T(t + s) = T(t)T(s) (t, s 2 0); •T€ C(0, 0), L(C")). Show that T is differentiable (from the right) in 0. Show that T(t) = e' A, where A = †(0). Hint: Show that v(t) := T(s) ds is invertible for small to and for those T(t) = V(to)(V(t +to) - V(t)).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Task 9 (Matrix Exponential I) Let T: [0, o0) + L(C") satisfy • T(0) = Idx; • T(t + s) = T(t)T(s) (t, s 2 0); •T€ C(0, 0), L(C")). Show that T is differentiable (from the right) in 0. Show that T(t) = e' A, where A = T(0). Hint: Show that V(t) := T(s) ds is invertible for small to and for those T(t) = V (to)(V (t + to) – V(t)).
Transcribed Image Text:Task 9 (Matrix Exponential I) Let T: [0, o0) + L(C") satisfy • T(0) = Idx; • T(t + s) = T(t)T(s) (t, s 2 0); •T€ C(0, 0), L(C")). Show that T is differentiable (from the right) in 0. Show that T(t) = e' A, where A = T(0). Hint: Show that V(t) := T(s) ds is invertible for small to and for those T(t) = V (to)(V (t + to) – V(t)).
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