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.