Linear Algebra
Dynamical Geometry Problem Sheet 6: Eigenvalues of Linear Transform
Sketch: Martti E. Pesonen, Problems: with Eric Lehman
Instructions.
The following JSP-applet constructions contain illustration of the eigenvalues
of linear mappings in the Euclidean plane R2.
Recall that the eigenvalues of a matrix
are the (real or complex) solutions of the characteristic equation
|
0 = |
det
| (A - lI) = (a - l)(d - l) - bc = l2 - (a+d)l+ (ad - bc), |
|
and by a slight simplification we get
|
(eig)(A) = l1,2 = |
1
2
|
|
æ
è
|
a+d ± |
| __________
Ö(a-d)2 + 4bc
|
ö
ø
|
. |
|
1. First Sketch: eigenvalues as functions of the matrix elements
The eigenvalues are represented
on the screen by a cyan and a green point.
Note that these points are on the horizontal axis when they are real.
In this sketch the linear transform is hidden behind a button.
Push button 'R' to reset, the red cross to clear traces.
Problems to the First Sketch
Here you are supposed to vary the matrix by sliding the brown and white points on the real line.
We start by investigating the behaviour of the roots of the chracteristic polynomial
l2 -
(a+d)l+ (ad -
bc) = 0.
1. Keep a, c and d fixed and change the value of b by moving the
corresponding white point.
a) For what value(s) b are the eigenvalues equal?
Find the answer experimentally on the screen and then by computation.
b) Explain the shape that appears on the screen when moving b.
2. Reset everything. Keep b, c and d fixed and change the value of a.
a) For what value(s) a are the eigenvalues equal?
Find the answer experimentally on the screen and then by computation.
b) Explain the shape that appears on the screen when moving a.
2. Second Sketch: eigenvalues of a linear transform
The eigenvalues are represented
on the screen by a blue and a green point.
Note that these points are on the horizontal axis when they are real.
In this sketch the linear transform is hidden behind a button.
Push button 'R' to reset, the red cross to clear traces.
Problems to the Second Sketch
Now you are shown a function L defined by the matrix A. Again you may
change the matrix elements by sliding the points on the real line.
1. Keep a, c and d fixed and change the value of b by moving the
corresponding white point.
a) For what value(s) b are the eigenvalues equal?
Find the answer experimentally on the screen and then by computation.
b) Explain the shape that appears on the screen when moving b ?
c) For what value(s) of b is it possible to choose the animation segment in such a way that its image
is parallel with it? Try first several values of b, for instance b = 4, b = 0, b = -3.
Give then the general rule.
2. Reset everything and keep b, c and d fixed and change the value of a.
Answer the same questions as above.
a)
b)
c)
3. Reset everything. Choose b = c = 3. Find a matrix A such that the image of the animation
circle is a circle.
What is its radius?
Does it remind you of someone?