Ch 7.6: Complex Eigenvalues
We consider again a homogeneous system of n first order
linear equations with constant, real coefficients,
x1  a11x1  a12 x2    a1n xn
x2  a21x1  a22 x2    a2 n xn

xn  an1 x1  an 2 x2    ann xn ,
and thus the system can be written as x' = Ax, where
 x1 (t ) 
 a11 a12



 x2 (t ) 
 a21 a22
x(t )  
, A







 x (t ) 
a
 n 
 n1 an 2
 a1n 

 a2 n 
  

 ann 
Conjugate Eigenvalues and Eigenvectors
We know that x = ert is a solution of x' = Ax, provided r is
an eigenvalue and  is an eigenvector of A.
The eigenvalues r1,…, rn are the roots of det(A-rI) = 0, and
the corresponding eigenvectors satisfy (A-rI) = 0.
If A is real, then the coefficients in the polynomial equation
det(A-rI) = 0 are real, and hence any complex eigenvalues
must occur in conjugate pairs. Thus if r1 =  + i is an
eigenvalue, then so is r2 =  - i.
The corresponding eigenvectors (1), (2) are conjugates also.
To see this, recall A and I have real entries, and hence
A  r1I ξ (1)  0
 A  r1I  ξ (1)  0  A  r2I ξ ( 2)  0
Conjugate Solutions
It follows from the previous slide that the solutions
x(1)  ξ (1) e r1t , x( 2)  ξ ( 2) e r2t
corresponding to these eigenvalues and eigenvectors are
conjugates conjugates as well, since
x( 2)  ξ ( 2) e r2t  ξ (1) e r2t  x (1)
Real-Valued Solutions
Thus for complex conjugate eigenvalues r1 and r2 , the
corresponding solutions x(1) and x(2) are conjugates also.
To obtain real-valued solutions, use real and imaginary parts
of either x(1) or x(2). To see this, let (1) = a + ib. Then
x (1)  ξ (1) e i t  a  ib e t cos  t  i sin  t 
 e t a cos  t  b sin  t   ie  t a sin  t  b cos  t 
 u(t )  i v(t )
where
u(t )  e t a cos  t  b sin  t , v(t )  e t a sin  t  b cos  t ,
are real valued solutions of x' = Ax, and can be shown to be
linearly independent.
General Solution
To summarize, suppose r1 =  + i, r2 =  - i, and that
r3,…, rn are all real and distinct eigenvalues of A. Let the
corresponding eigenvectors be
ξ (1)  a  ib, ξ ( 2)  a  ib, ξ (3) , ξ ( 4) ,, ξ ( n )
Then the general solution of x' = Ax is
x  c1u(t )  c2 v(t )  c3ξ (3) e r3 t    cnξ ( n ) e rn t
where
u(t )  e t a cos  t  b sin  t , v(t )  e t a sin  t  b cos  t 
Example 1: (1 of 7)
Consider the homogeneous equation x' = Ax below.
1 
 1/ 2


x
x 
 1 1/ 2 
Substituting x = ert in for x, and rewriting system as
(A-rI) = 0, we obtain
1
 1/ 2  r
 1   0 

    
 1 / 2  r  1   0 
 1
Example 1: Complex Eigenvalues
(2 of 7)
We determine r by solving det(A-rI) = 0. Now
1/ 2  r
1
1
1/ 2  r
 r  1 / 2  1  r 2  r 
2
5
4
Thus
 1  12  4(5 / 4)  1  2i
1
r

  i
2
2
2
Therefore the eigenvalues are r1 = -1/2 + i and r2 = -1/2 - i.
Example 1: First Eigenvector
(3 of 7)
Eigenvector for r1 = -1/2 + i: Solve
1
 1/ 2  r
 1   0 
    
 
 1 / 2  r  1   0 
 1
1 1   0 
i  1   0 
i
 1
      
    
 
  1  i   2   0 
  1  i   2   0 
A  rI ξ  0
by row reducing the augmented matrix:
i 0  1 i 0
  i 2 
 1
1
(1)
(1)
  choose ξ   

  
  ξ  
 1  i 0   0 0 0 
i
 2 
Thus
ξ
(1)
 1  0 
    i  
 0   1
Example 1: General Solution (5 of 7)
The corresponding solutions x = ert of x' = Ax are
u(t )  e
t / 2
v(t )  e
t / 2
 1

 0
t / 2  cos t 

  cos t    sin t   e 
 1
  sin t 
 0 

 1

0
t / 2  sin t 

  sin t    cos t   e 
 1
 cos t 
 0 

The Wronskian of these two solutions is

(1)
W x ,x
( 2)
(t ) 
e t / 2 cos t
e t / 2 sin t
 e t / 2 sin t e t / 2 cos t
 e t  0
Thus u(t) and v(t) are real-valued fundamental solutions of
x' = Ax, with general solution x = c1u + c2v.