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final_review (dragged) 5 - 6 MA 36600 FINAL REVIEW 7.7...

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6 MA 36600 FINAL REVIEW § 7.7: Fundamental Matrices. Consider the initial value problem d dt x = P ( t ) x , x ( t 0 ) = x 0 . Say that x (1) , x (2) , . . . , x ( n ) is a fundamental set of solutions. Then the general solution is x ( t ) = c 1 x (1) ( t ) + c 2 x (2) ( t ) + · · · + c n x ( n ) ( t ) = Ψ ( t ) c in terms of the matrices Ψ ( t ) = x 11 ( t x 12 ( t ) · · · x 1 n ( t ) x 21 ( t ) x 22 ( t ) · · · x 2 n ( t ) . . . . . . . . . . . . x n 1 ( t ) x n 2 ( t ) · · · x nn ( t ) and c = c 1 c 2 . . . c n . We call Ψ ( t ) = x ij ( t ) a fundamental matrix . It satisfies the matrix equation d dt Ψ = P ( t ) Ψ . Since the Wronskian is W = det Ψ , we see that Ψ ( t ) is a nonsingular matrix. Hence we can choose the constant c in order to solve the initial value problem: x 0 = Ψ ( t 0 ) c = c = Ψ ( t 0 ) 1 x 0 . Say P ( t ) = A is an n × n constant matrix. The solution to the initial value problem is x ( t ) = Φ ( t ) x 0 in terms of the matrix exponential Φ ( t ) = exp ( A t ) = k =0 A k t k k ! = I + A t + A 2 t 2 2 + · · · . In general, the initial value problem d dt Ψ = A Ψ , Ψ (0) = Ψ 0 has the unique solution Ψ ( t ) = exp ( A t ) Ψ 0 . The order does matter! The matrix Ψ 0 exp ( A t ) is not a solution to the initial value problem. We say that an n × n matrix is a
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