c36 - 18.03 Class 36 May 10 Review of matrix exponential...

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18.03 Class 36, May 10 Review of matrix exponential Inhomogeneous linear equations [1] Prelude on linear algebra: If A and B are matrices such that the number of columns in A is the same as the number of rows in B , then we can form the "product matrix" AB. If the columns of B are b_1 , ... , b_n then the columns of AB are A b_1 , ... , A b_n [2] I know you don't want to hear more about Romeo and Juliet. This is about amorous armadillos, named Xena and Yan. x' = - x + 3y y' = -3x - y Matrix [ -1 , 3 ; -3 , -1 ] . Characteristic poly: lambda^2 + 2 lambda + 10 Eigenvalues: -1 +- 3i Stop! what do we want to know? We have a stable spiral, rotating clockwise. Do we want more? We already know that this romance will peter out into dull acceptance. Maybe that's enough information about X and Y's love life for us. If not, we can go ahead and solve: Eigenvector for -1 + 3i : [3 ; 3i] or just [1;i] . Normal mode: e^{(-1+3i)t} [1;i] Basic real solutions: e^{-t}[cos(3t) ; -sin(3t)] e^{-t}[sin(3t) ; cos(3t)] Fundamental matrix Phi(t) = e^{-t} [ cos(3t) , sin(3t) ; -sin(3t) , cos (3t) ] Then the general solution is Phi(t) [ a ; b ] .
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