Dt 0 233 for students with access to mathematica a

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Unformatted text preview: the Spectral Theorem of Section 2.1 guarantees that the eigenvalues of A are real and that we can find an n × n orthogonal matrix B such that λ1 0 · · · 0 0 λ2 · · · 0 . B −1 AB = · · · 0 0 · · · λn If we set x = B y, then B dy dx = = Ax = AB y dt dt ⇒ dy = B −1 AB y. dt Thus in terms of the new variable y1 dy1 /dt λ1 y2 dy2 /dt 0 y = , we have · · = · dyn /dt 0 yn 0 λ2 · 0 ··· ··· ··· y1 0 0 y2 , · · λn yn so that the matrix differential equation decouples into n noninteracting scalar differential equations dy1 /dt = λ1 y1 , dy2 /dt = λ2 y2 , · dyn /dt = λn yn . 45 We can solve these equations individually, obtaining the general solution λ t c1 e 1 y1 y2 c2 eλ2 t y= = · · , yn cn eλn t where c1 , c2 , . . . , cn are constants of integration. Transferring back to our original variable x yields λ t c1 e 1 c2 eλ2 t λ1 t λ2 t λn t x=B · = c1 b1 e + c2 b2 e + · · · + cn bn e , cn eλn t where b1 , b2 , . . . , bn are the columns of B . Note that x(0) = c1 b1 + c2...
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