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final_review (dragged) 4

# final_review (dragged) 4 - MA 36600 FINAL REVIEW 5 7.5...

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MA 36600 FINAL REVIEW 5 § 7.5: Homogeneous Linear Systems with Constant Coe cients. Consider the homogeneous linear system with constant coe cients d dt x = A x . We may guess a solution in the form x ( t ) = ξ e rt for some constant vector ξ and constant scalar r . (The symbol “ ξ ” is the letter “xi,” and is pronounced “zai” or “ksi.”). Then ( A r I ) ξ = 0 , so that r is an eigenvalue of A and ξ is an eigenvector of A . § 7.6: Complex Eigenvalues. Consider a 2 × 2 matrix A = a 11 a 12 a 21 a 22 = det ( A r I ) = r 2 (tr A ) r + (det A ) in terms of the trace and determinant of A : tr A = a 11 + a 22 , det A = a 11 a 22 a 12 a 21 . The discriminant of this polynomial is disc A = (tr A ) 2 4 (det A ) = ( a 11 a 22 ) 2 + 4 a 12 a 21 . If the discriminant is positive, then the eigenvalues r are real. If the discriminant is negative, then the eigenvalues r are imaginary. We consider a homogeneous system d dt x = A x where A is a 2 × 2 matrix with constant coe cients with disc A < 0. Its eigenvalues r and eigenvectors
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