This distance is the "obvious" singular value version of the class of Finsler metrics on the cone of positive definite matrices. A basic paper (not the earliest, but one with useful references that'll be handy) to start off is: [On the exponential metric increasing property] 1 by R. Bhatia.
Once I get back to my computer, I'll try tracking down some more info, especially, that with a longer historical perspective.