Some more little titbits about the eigen-family:
is a line through the origin
on which the eigenvectors fall.
i.e. - For an eigenvector
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
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
Ax = [a b] [x] = [ax + by]
[c d] x [y] [cx + dy]
kx = k[x] = [kx]
[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
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.