Graphics Reference
In-Depth Information
Let's derive the characteristic equation (
4.18
).
Consider the 2D transform
t
that maps the point
(x, y)
to another point
(ax
+
by, cx
+
dy)
:
t (x,y)
→
(ax
+
by, cx
+
dy) .
This is expressed in matrix form as
t
:
v
→
Av
or
x
y
ab
cd
x
y
=
where
ab
cd
,
x
y
.
A
=
v
=
Therefore, if
v
is an eigenvector of
t
, and
λ
its associated eigenvalue, then
=
Av
λ
v
ab
cd
x
y
λ
x
y
=
or in equation terms:
ax
+
by
=
λx
cx
+
dy
=
λy.
Rearranging, we have
(a
−
λ) x
+
by
=
0
cx
+
(d
−
λ) y
=
0
or back in matrix form:
a
x
y
0
0
.
−
λ
b
=
cd
−
λ
T
For a non-zero
[
xy
]
to exist, we must have
det
a
−
λb
=
0
−
c
d
λ
which is called the
characteristic equation
. Let's use this on the transform
41
14
x
y
x
y
.
=
