Complex Eigenvalues
26 Nov 2012
“The circle is now complete” – Darth Vader
As if real eigenvalues weren’t complicated enough …
We’ve already seen that a singular matrix has a 0 eigenvalue:
How about a symmetric singular matrix?
These examples remind us of two very useful properties:
 The trace of a matrix is the sum of its eigenvalues
 The determinant of a matrix is the product of its eigenvalues.
In this 2x2 case, we are finding roots of the characteristic polynomial:
Thus for a 2x2 matrix, the characteristic polynomial (cp) is  2  tr[ A]  det[ A]  0 and
that might just have complex roots!
Using the quadratic equation, we can form the discriminant (that old B2 - 4AC thing),
disc[cp]  tr 2 [ A]  4 det[ A] , which will be negative for complex roots. This gives the
condition
tr 2 [ A]  4 det[ A]
(a  d ) 2  4(ad  bc)
,
a 2  2ad  d 2  (a  d ) 2  4bc
How does that apply to the theorem “symmetric matrices have real eigenvalues”?
Examples
1 1
B

1 1 
1  i 
det[ B ]  2 tr[ B ]  2   

1  i 
 2 1
C

1 3 
det[C ]  7 tr[C ]  5  
1 5  i 3 


2 5  i 3 
Note that complex eigenvalues are always conjugate pairs x  iy , so that both the trace
and determinant properties still hold. Recall: addition of complex conjugates adds their
real parts: ( x  iy )  ( x  iy )  2 x ; multiplication of complex conjugates adds the squares
of their real and imaginary parts: ( x  iy )( x  iy )  x 2  y 2 . This is just the square of the
magnitude of the vector {x, +/-y} represented by the complex number x  iy .
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The complex eigenvalue calculation can be generalized to   (tr[ A]  i  ) , where
2
2
  4 det[ A]  (tr[ A]) . If  > 0, we have complex eigenvalues.
Imagine this … complex eigenvectors too!
With complex eigenvalues, the eigenvectors must also have complex components: In the
first example above, the eigenvectors may be found by solving the equations that result
from
( B  1 I ) x  0
1   x  0 
1  (1  i )

 1
1  (1  i )   y  0 

1
i 
y  ix and x  iy, ie   or   , etc
 i 
1
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Find the eigenvector corresponding to the conjugate eigenvalue; find the eigenvectors of
matrix C in the second example above. Do these by hand (check with Eigenvectors[ ]).
Go back to the applet from L3-3,
http://www.math.duke.edu/education/webfeatsII/Lite_Applets/Eigenvalue/applet.html
and enter a matrix which you know has complex eigenvectors (such as one of those
above). Try to drag the red square to find an eigenvector – if you can!
What do these complex eigenvalues mean?
Suppose you are diabetic: You must take insulin in order to regulate your blood sugar.
At any time, the “excess” glucose and insulin levels in your blood stream can be modeled
by the matrix equation
 g n 1  .9 .4   g n 
 i   .1 .9   i  This is known as a discrete dynamic system.
 n 
 n 1  
This is similar to our work with population models and other Markov chains (Lab 1-10):
A vector representing the current state of the system is multiplied by a transition matrix,
which details how the system changes from one time to the next. Given an initial state,
this equation is evaluated multiple times (which is just like raising the transition matrix to
an integer power).
The plot of {excess glucose,
insulin} pairs obtained with
this transition matrix results in
a counter-clockwise spiral.
Initial point
This system converges slowly from the initial point (for example, {100, 0} representing
your blood sugar immediately after a meal) towards your baseline (fasting) level of{0, 0}.
Your excess glucose level oscillates from positive to negative (meaning your blood sugar
might drop dangerously low), but at least the amplitude of oscillation decreases with time
What are the eigenvalues of the transition matrix?
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Compare the graph above to that
of a stochastic (columns sum to 1)
transition matrix:
 g n 1  .9 .1  g n 
 i   .1 .9   i 
 n 
 n 1  
What are the eigenvalues of this
transition matrix?
Problem
Find a transition matrix for this
system that fails the  test
above, but does not oscillate
(similar to the plot shown).
How could you predict whether
the system oscillates or not?
Why are complex eigenvalues associated with oscillatory behavior?
cos   sin  
Consider a rotation matrix R  
 ; we have tr[R] = 2 cos  and det[R] = 1.
 sin  cos  
Thus   4(1  cos 2  )  4sin 2   0 ; the eigenvalues of the rotation matrix are complex
except when  = 0, , 2, etc. Using the eigenvalue formula above, we can write
  cos   i sin  , which you may recognize as   e
 i
. Successive rotations from
an initial point on the unit circle take us around and around – so our x and y components
oscillate from +1 to – 1 and back.
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Treating the complex eigenvalue as a vector, we can write    cos 
  arctan
sin   , so that
Im[ ]

 arctan
. What is the magnitude of this vector?
Re[ ]
tr[ R]
i 
 1
Show that the eigenvectors of the rotation matrix are v    or   and that these
1
i 
eigenvectors are independent of the rotation angle.
Example
 2 1
Let A  
 , which is a rotation with stretch ( det[ A]  1 ).
3 2 
The characteristic polynomial of A is  2  4  7  0 and therefore the eigenvalues are
 x1 
 2 1  x1 
2  i 3 . Find the corresponding eigenvectors by solving 
(2
3)
i


y 
 
 3 2   y1 
 1
x 
 2 1  x2 
and 
 (2  i 3)  2  .



 3 2   y2 
 y2 
Rotational magic
 1 
One choice for an eigenvector for the matrix A used above is v1  
 , which we can

i
3


1   0 
write in a + ib form v1     i 
 . Use the Im and Re parts of this eigenvector to
0    3 
1
 0
form the matrix (conveniently, a basis for the real-imaginary plane) P  
 and

3
0


find its inverse (which is also its transpose – you did notice that the columns are
orthogonal?)
Now consider the product
1
| |
( PT AP) , where |  | Re2 ( )  Im 2 ( )  7 .
cos   sin  
Incredibly, this product results in a rotation matrix of the form R  
!
 sin  cos  
There’s no stretch, as we’ve already seen that the determinant of any such matrix is 1.
We should be able to take any matrix with complex eigenvectors, form a basis of
orthogonal vectors and produce such a rotation matrix. The angle of rotation can be
found from any of the values of R.
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Find the angle of rotation represented by the above matrix A. Then plot the result of
applying the linear transform Ax to a series of points on the unit circle. What is the shape
of the image?
Problem
Find a matrix that transforms the unit circle into an ellipse with major axis of length 2
x2 y 2
and minor axis of length 1 (recall that an ellipse is 2  2  1 , where the major axis has
a
b
length 2a and the minor axis length 2b). Write your matrix in the form |  | R  PT AP
by following the procedure outlined above.
If you’ve never seen complex eigenvalues in the context of differential equations, refer to
Dawkins’ DE chapter:
http://tutorial.math.lamar.edu/Classes/DE/ComplexEigenvalues.aspx
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