MA 36600 FINAL REVIEW5§7.5: Homogeneous Linear Systems with Constant Coeﬃcients.•Consider the homogeneous linear system with constant coeﬃcientsddtx=A x.We may guess a solution in the formx(t) =ξertfor some constant vectorξand constant scalarr.(The symbol “ξ” is the letter “xi,” and is pronounced “zai” or “ksi.”). Then (A−rI)ξ=0, so thatris an eigenvalue ofAandξis an eigenvector ofA.§7.6: Complex Eigenvalues.•Consider a 2×2 matrixA=a11a12a21a22=⇒det (A−rI) =r2−(trA)r+ (detA)in terms of thetraceanddeterminantofA:trA=a11+a22,detA=a11a22−a12a21.The discriminant of this polynomial isdiscA= (trA)2−4 (detA) = (a11−a22)2+ 4a12a21.If the discriminant is positive, then the eigenvaluesrare real. If the discriminant is negative, thenthe eigenvaluesrare imaginary.•We consider a homogeneous systemddtx=A xwhereAis a 2×2 matrix with constant coeﬃcients with discA<0. Its eigenvaluesrand eigenvectors
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