Take the tour ×  In this thesis, Pedro Freitas discusses the
properties of distance functions on matrices defined by $d_p(A,
B) = (\sum (\log (\sigma_i(A^{-1} B)))^p)^{1/p}.$ Here
$\sigma_i$ are the singular values of the argument. This is
presumably standard, and goes back to Minkowski, but I am having
trouble finding references or any kind of comprehensive
treatment. Any suggestions for where this might be discussed?
(number theory tag because this is closely related to the Siegel
half space, etc).  share|improve this question add comment  1
Answer  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 by R. Bhatia.  Once I get back to my
computer, I'll try tracking down some more info.  EDIT. The
three references cited below [1-3] will provide a deeper
historical perspective. I am including