In the last video, we talked about
what happens when you diagonalize
an arbitrary unitary operator.
In this video, we're going to talk
about the example that
motivated the whole subject,
which is rotations in good old
three dimensional space.
So first of all, let's look at the
example of rotating things
about the z axis.
So we rotate in the x-y plane,
we leave the z axis alone.
Now we've seen that the eigenvalues
of this 2x2 matrix are cos(θ)+isin(θ)
and cos(θ)-isin(θ), in other words,
e^{iθ} and e^{-iθ}.
And the third eigenvalue is one,
that's what you get from
the lower right hand corner.
So those are our eigenvalues and
the eigenvectors, the eigenvector
with eigenvalue one is 001 and
that gives us our axis of rotation.
The eigenvector with eigenvalue e^{iθ}
is i10, and if you look at
the imaginary part of that, that's 100
and the real part is 010, and if
you take the cross product of the
imaginary part with the real part,
so e_1 cross e_2, you get e_3,
which is the axis of rotation.
This is the signature of a counter-
clockwise rotation.
The imaginary part of the e^{iθ}
eigenvector cross the real part
gives you the axis.
Finally, the trace, well you just add
up the diagonal entries.
cos(θ)+cos(θ)+1, that's
1+2cos(θ).
It's also the sum of the eigenvalues,
1+e^{iθ}+e^{-iθ}.
Okay, so now let's switch gears and go
to a general rotation, a rotation about
any old axis, and the first thing to
notice is that the axis itself
is always an eigenvector with
eigenvalue one because no matter
what direction you're rotating around,
the axis itself doesn't move.
If we're rotating in the x-y plane,
then the z axis doesn't go anywhere.
Also, if the angle of rotation is θ,
we've already seen that rotating by θ
gives you eigenvectors which are
e^{iθ} and e^{-iθ}.
So if you ever have a rotation by θ
about some axis, then your
eigenvalues are going to be one for
the axis, e^{iθ} and e^{-iθ}.
And that means the trace is going
to be 1+2cos(θ), which means
you can recover θ from the trace.
Take the trace minus one,
divided by two, take the inverse
cosine.
Now that gives you the angle of rotation,
but it doesn't tell you which way
you're rotating, clockwise
or counterclockwise.
And that's a matter of perspective.
If you look at a car that's driving
down the street, are its wheels turning
clockwise or counterclockwise?
Well, that depends on where you're
looking at the car from.
If you're looking from the driver's
side, you're going to say that
the wheels are turning counterclockwise.
If you're looking from the passenger's
side, you're going to say they're
turning clockwise.
Also, if you have a matrix and its
inverse matrix, they're both
going to be rotations. If this is
a counterclockwise rotation,
this is a clockwise rotation. If this
is clockwise, this is counterclockwise.
Yet they had the exact
same eigenvalues.
The eigenvalues of R inverse are
the reciprocals of the eigenvalues of R,
but the reciprocal of one is one,
the reciprocal of e^{iθ} is e^{-iθ},
and the reciprocal of e^{-iθ} is e^{iθ}.
So you can't tell whether it's clockwise
or counterclockwise by looking
at the eigenvalues, you have to
look at the eigenvectors.
You have to look at the eigenvector
with eigenvalue e^{iθ},
take it's imaginary part cross the real
part, and if that's pointing
in the direction of your axis,
it's counterclockwise.
If it's pointing opposite the direction
of your axis, then it's clockwise.
That's a subtle point, not a big deal
if you get the details of it.
The important thing is to find the
angle of rotation.
So let's look at this example.
Here's matrix and you can see that
all three columns are orthonormal
In fact, they're all just permutations
of each other, you just have
two 2/3 and one -1/3, and sure
enough, the cross product
of the first two is the third, cross
product of the second and the third
is the first, cross product of the third
and the first is the second.
The determinant is plus one and
they're orthonormal columns,
so this is an orthogonal matrix,
it's got to be a rotation.
If it's a rotation, we should be able
to find the axis and find the angle.
So the angle is easy because the trace
is 2/3+2/3+2/3, that's two.
So the cos(θ) has to be the trace
minus one over two, which is a 1/2.
Since the cosine is 1/2, θ has to be
pi/3, which is 60°.
This is a 60° rotation about some axis.
How do we find the axis, we find
the eigenvector with eigenvalue one
and we do that by row reducing R-I.
So there's R-I, you row reduce it,
and you get that the eigenvector is
111, or if you want a normalized
eigenvector, it's 111 over √3.
In fact, you can see that if you
multiply this by 111, you get 111.
The sum of the entries in the first
row is one, the sum of the entries
in the second row is one, sum of
the entries of the third row is one,
so 111 is an eigenvector with
eigenvalue one.
Now our second example has the same
columns as the first example
except I took this column and I moved
it to the second slot and took
this column and I moved it to the
third slot, and took this column
and I moved it to the first slot.
I just permuted the columns
and that has the same trace, it's got
a trace of 2, and in fact,
it's the same axis, the same angle,
it's just going to the other direction.
That means that it must be the
inverse of the first matrix and it is.
It's the transpose of the first matrix.
This is the transpose of this, it's
an orthogonal matrix so
the transpose is the inverse.
So the inverse of rotating counter-
clockwise is rotating clockwise.
And our last example, again the same
columns in a different order,
only now the trace is -1. If the trace
is -1, then the cosine is (-1-1)/2
which is -1, which means that θ
has to be pi.
So this is a rotation by 180° about
the same 111 axis.
And how is a rotation by 180° related
to rotation by 60°? Well if you do
a rotation by 60° three times,
that's a rotation by 180.
And if you rotate by 60° the other
way three times, that's also
a rotation by 180.
So without doing a whole lot of
work, just by looking at traces,
by traces and finding a single
eigenvector, we can find out
an awful lot about a matrix.
