This should be easy, but...
I would like to express the singular value decomposition of a 2 x 2 complex matrix $A$ as function of its coefficients $A_{ij}$. In "closed form", no intermediate values, straight up.
What I mean is that if I express the SVD in terms of eigenvalues, say, then I have to write those eigenvalues in terms of the $A_{ij}$'s and substitute. In other words, I don't want to write the solution to SVD in terms of the solution of something else.
Possibly useful background:
In the Wikipedia page on SVD, if you write $A$ as a linear combination of the Pauli matrices (and identity), the singular values are expressed in terms of the corresponding coefficients. But then I don't know how you find those coefficients, so I can't tell if they themselves are in "closed form".
http://en.wikipedia.org/wiki/Singular_value_decomposition
Thanks.