# Worked Examples: Complex Eigenvalues

## A Two Dimensional Rotation

The matrix that represents an anticlockwise rotation by degrees is

with respect to the standard basis.
If

then

and

Therefore

which we can factorise as

and hence

is an eigenvector with eigenvalue and

is an eigenvector with eigenvalue .

## A Two Dimensional Rotation then Magnification

The matrix that represents an anticlockwise rotation by degrees followed by a magnification of is

with respect to the standard basis.
If

then

and

so

which factorises as

Hence

is an eigenvector with eigenvalue and

is an eigenvector with eigenvalue .

## A Three Dimensional Rotation

The matrix that represents an anticlockwise rotation by degrees around the vector

is

with respect to the standard basis.
Since we spot that the sum of the second and third columns has a particularly nice form we try the initial vector

and see that

and so

is an eigenvector with eigenvalue .
(This shows us the axis of rotation.)
If

then

and

hence

factorising gives

so

is an eigenvector with eigenvalue .
Also

is an eigenvector with eigenvalue .