MA 36600 FINAL REVIEW 5 § 7.5: Homogeneous Linear Systems with Constant Coeﬃcients. • Consider the homogeneous linear system with constant coeﬃcients d dt x = Ax . 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 = Ax where A is a 2 × 2 matrix with constant coeﬃcients with disc A < 0. Its eigenvalues r and eigenvectors ξ are imaginary. We know that one solution of the system is
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.