Complex Eigenvalues

Example 1: Direction Field(1 of 7)* Consider the homogeneous equation x'= Ax below-1/2-1 -1/2)#Adirection field for this system is given below Substituting x = Eert in for x, and rewriting system as(A-rDE=0,we obtain(-1/2-r(OY5心-1 -1/2-r八）（0

Example 1: Direction Field (1 of 7) Consider the homogeneous equation x' = Ax below. A direction field for this system is given below. Substituting x = e rt in for x, and rewriting system as (A-rI) = 0, we obtain x x        − − −  = 1 1/ 2 1/ 2 1         =                − − − − − 0 0 1 1/ 2 1/ 2 1 1 1   r r

Example 1:Complex Eigenvalues(2of7)* We determine r by solving det(A-rD) = 0. Now-1/2-r15=(r+1/2) +1=r2 +r+-14-1/2-rThus装5-1± /12 -4(5 /4)1-1±2iir=222* Therefore the eigenvalues are ri =-1/2 + i and r2 = -1/2 - i

Example 1: Complex Eigenvalues (2 of 7) We determine r by solving det(A-rI) = 0. Now Thus Therefore the eigenvalues are r1 = -1/2 + i and r2 = -1/2 - i. ( ) 4 5 1/ 2 1 1 1/ 2 1/ 2 1 2 2 = + + = + + − − − − − r r r r r i i r = −  −  = −  − = 2 1 2 1 2 2 1 1 4(5/ 4) 2