Some more little titbits about the eigen-family:
The
eigenline is a line through the
origin on which the eigenvectors fall.
i.e. - For an
eigenvector
[x]
[y]
The eigenline is a line ax+by=0, where a and b are
constants, that satisfies the values of x and y.
e.g. - For the eigenvector
[ 1 ]
[ 2 ]
The eigenline is 2x-y=0
A handy way of finding the eigenvalues of any 2x2
matrix A:
[a b]
[c d]
Is to solve the
characteristic equation:
k^2 - (a+d)k + ad - bc = 0 (1)
To find any eigenvalues k. If there are no
real solutions to this equation, there are no eigenvalues.
For each eigenvalue k found from solving (1), substitute it into the eigenvalue equation Ax = kx, where x is the
vector
[x]
[y]
Ax = [a b] [x] = [ax + by]
[c d] x [y] [cx + dy]
kx = k[x] = [kx]
[y] [ky]
[ax + by] = [kx]
[cx + dy] [ky]
Solving the
simultaneous equations
ax + by = kx
cx + dy = ky
For an eigenvalue k gives us an equation of the eigenline of the form
mx + ny = 0
And the eigenvector is any pair of values which satisfy the equation of the eigenline.
As the number of eigenvalues is given by the
roots of the
quadratic (1), the number can be found from the
discriminant of a quadratic equation:
b^2 - 4ac
In (1)
a = 1
b = a+d
c = ad - bc
So the discriminant comes out as:
(a+d)^2 - 4(ad-bc)
After a brief bit of juggling, we can arrive at the following conclusions:
If a=d, the discriminant boils down to bc, ie:
bc < 0 There are no eigenvalues.
bc = 0 There is 1 eigenvalue.
bc > 0 There are 2 eigenvalues.
If a ≠ b, there are either 0 or 2 eigenvalues.