# Worked Examples: Complex Eigenvalues

## A Two Dimensional Rotation

The matrix that represents an anticlockwise rotation by $45$ degrees is

with respect to the standard basis. If

then

and

Therefore

which we can factorise as

and hence

is an eigenvector with eigenvalue $\frac{\sqrt{2}}{2}(1-i)$ and

is an eigenvector with eigenvalue $\frac{\sqrt{2}}{2}(1+i)$.

## A Two Dimensional Rotation then Magnification

The matrix that represents an anticlockwise rotation by $90$ degrees followed by a magnification of $2$ is

with respect to the standard basis. If

then

and

so

which factorises as

Hence

is an eigenvector with eigenvalue $-2i$ and

is an eigenvector with eigenvalue $2i$.

## A Three Dimensional Rotation

The matrix that represents an anticlockwise rotation by $90$ 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 $1$. (This shows us the axis of rotation.) If

then

and

hence

factorising gives

so

is an eigenvector with eigenvalue $-i$. Also

is an eigenvector with eigenvalue $i$.