This thesis states
... the $L^2$ or trace norm, [is] defined to be $||M|| = Tr \sqrt{M M^\dagger}$. If elements of SU(2) are associated with points on the surface of the 3-sphere, then this is simply equivalent to twice the Euclidean distance.
I have two points of confusion. The first is the definition of $L^2$ when applied to elements of SU(2). I have read (here for instance) that for normal matrices, this is equivalent to the sum of the eigenvalues of $M$. But for SU(2), because we have unitary matrices, does this not become $Tr \sqrt{M M^\dagger} = Tr \sqrt{I_2} = Tr(I_2) = 2$? Is it the case that all of the trace norms of matrices in SU(2) are then equal? This doesn't seem to match up with my calculation below using eigenvalues, so I assume I have made a mistake somewhere.
Putting that aside for a moment, I seem to be able to calculate the eigenvalues of an arbitrary matrix in SU(2). After a bit of calculation, the characteristic polynomial can be written as
$$ \lambda^2 - 2 \Re(z) \lambda + 1 $$
where $z$ is the parameter along the main diagonal. This gives the eigenvalues $\Re(z) \pm \sqrt{ \Re(z)^2 - 1 }$, and seems to match up with a couple of manual calculations I did for particular matrices. My questions:
- Is it correct that the trace norm is then $\sum \left( \Re(z) \pm \sqrt{ \Re(z)^2 - 1 } \right) = 2 \Re(z)$?
- This seems suspect, because if I view elements of SU(2) as unit quaternions, these eigenvalues are in terms of just the real part. If correct, is there some intuition about why this is the case?
- If correct, is this enough to show the claim above about Euclidean distance? I'm not seeing the connection immediately
Edit:
On a second look, I think that the expression in the first bullet should be sum of the norms of the eigenvalues:
$$ \sum \left | \Re(z) \pm \sqrt{ \Re(z)^2 - 1 } \right | = 2 \left | \Re(z) \pm \sqrt{ \Re(z)^2 - 1 } \right | $$
which gives me the expected trace norm of 2 for the few SU(2) matrices I checked by hand. So the calculation makes sense, though the intuition of the 3-sphere and interpretation as a function of the real component of a quaternion is still unintuitive to me.