c33 - 18.03 Class 33, May 3 Complex or repeated eigenvalues...

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18.03 Class 33, May 3 Complex or repeated eigenvalues [1] The method for solving u' = Au that we devised on Monday is this: (1) Write down the characteristic polynomial p_A(lambda) = det(A - lambda I) = lambda^2 - (tr A)lambda +(det A) (2) Find its roots, the eigenvalues lambda_1, lambda_2 (3) For each eigenvalue find a nonzero eigenvector --- v such that Av = lambda v or (A - lambda I) v = 0 --- say v_1 , v_2. Then the "ray" solutions are multiples of e^{lambda_1 t} v_1 and e^{lambda_2 t} v_2 These are also called "normal modes." The general solution is a linear combination of them. [2] This makes you think there are always ray solutions. But what about the Romeo and Juliet example, which spirals and obviously has no such solution? Or, what about A = [ 1 2 ; -2 1 ] . I showed the trajectories on Linear Phase Portraits: Matrix Entry. Let's apply the method and see what happens. tr(A) = 2 , det(A) = 5, so p_A(lambda) = lambda^2 - 2 lambda + 5 which has roots lambda_1 = 1 + 2i, lambda_2 = 1 - 2i. (As always for real polynomials, the roots (if not real) come as complex conjugate pairs.) We could abandon the effort at this point, but we had so much fun and success with complex numbers earlier that it seems we should carry on. Find an eigenvector for lambda_1 = 1 + 2i : A - (1+2i)I : [ - 2i , 2 ; -2 , -2i ][ ? ; ? ] = [ 0 ; 0 ] Standard method: use the entries in the top row in reverse order with one sign changed: [ 2 ; 2i ] or, easier, in this case,
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c33 - 18.03 Class 33, May 3 Complex or repeated eigenvalues...

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