I'm working to have a greater understanding of the arbitrary unitary transformation matrix when working in the context of the Bloch sphere. At this time I have found several equivalent representations of this arbitrary unitary, but I'm trying to bridge the gap between a version that I feel I understand well, and another that I'm not yet clear on why it is different.
The one that I feel I understand well is $$U_1=\exp(i\gamma)\exp(-i\alpha/2 \hat{n}\cdot\vec{\sigma})$$ which I found and worked through with the help of this set of notes from Ian Glendinning.
The second version is from Audretch's Entangled Systems; New Directions in Quantum Physics. I do not have a link for this book, but on Page 56 he gives the following version of the unitary with little explanation as to its derivation. $$U_2=\exp(i\kappa)\exp(-i\lambda/2\sigma_z)\exp(-i\mu/2\sigma_y)\exp(-i\nu/2\sigma_z)$$
I know that $\kappa$ and $\gamma$ are just different names for the global phase factor.
I believe that $\alpha$ and $\mu$ represent the arbitrary angle of rotation around the $\hat{n}$ axis.
I believe the differences relate primarily to the fact that Audretsch's representation is representing the pairwise orthonormal columns and rows as discussed in Audretsch's book as well as in this post: General parametrisation of an arbitrary $2\times 2$ unitary matrix?)
Any help I could get in better understanding how these are equivalent representations would be appreciated.
With the help of Wikipedia and the stack exchange discussion about general parametrisation linked above, I have been able to fully understand the origin of $U_2$ and how to derive it from an arbitrary 2 x 2 matrix. I am still trying to equate this with $U_1$ above.