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MIT18_03S10_c36

# MIT18_03S10_c36 - 18.03 Class 36 May 5 2010 The matrix...

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18.03 Class 36 , May 5, 2010 The matrix exponential: initial value problems. 1. Definition of e^{At} 2. Computation of e^{At} 3. Uncoupled example 4. Defective example 5. Exponential law [1] Recall from day one: (a) x' = rx with initial condition x(0) = 1 has solution x = e^{rt} . x' = rx with any initial condition has solution x = e^{rt} x(0) . Later, we decided to *define* e^{it} as the solution to (b) x' = ix with initial condition x(0) = 1 . Following Euler, a solution is given by cos t + i sin t , so we found that e^{it} = cos(t) + i sin(t) (c) Now we are studying u' = A u . Let's try to *define* The solution to u' = Au with initial condition u(0) is u = e^{At}u(0). (**) Note that the initial value u(0) is a vector, and u(t) is a vector valued function. So the expression e^{At} must denote a matrix, or rather a matrix valued function. What could e^{At} be? For a start, what is its first column? Recall that the first column of any matrix B is the product B[1;0] , and [ b_1 b_2 ] [1;0] = b_1 , so combining this with (**) we see: The first column of e^{At} is the solution to u' = Au with u(0) = [1;0]. Similarly, The second column of e^{At} is the solution to u' = Au with u(0)= [0;1].

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MIT18_03S10_c36 - 18.03 Class 36 May 5 2010 The matrix...

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