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Task 9 (Matrix Exponential I) Let T : [0, 0) –→ L(C") satisfy • T(0) = Idx; T(t + s) = T(t)T(s) (t, s > 0); • TE 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) := f 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, 0) –→ L(C") satisfy • T(0) = Idx; T(t + s) = T(t)T(s) (t, s > 0); • TE 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) := f T(s) ds is invertible for small to and for those T(t) = V-'(to)(V(t + to) – V (t)).

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 70EQ
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Task 9 (Matrix Exponential I) Let T : [0, 00) → L(C") satisfy • T(0) = Idx; • T(t+ s) = T(t)T(s) (t, s > 0); • TE C([0, 0), L(C")). Show that T is differentiable (from the right) in 0. Show that T(t) = e'A, whcre 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, 00) → L(C") satisfy • T(0) = Idx; • T(t+ s) = T(t)T(s) (t, s > 0); • TE C([0, 0), L(C")). Show that T is differentiable (from the right) in 0. Show that T(t) = e'A, whcre 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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