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.

**EDIT.** The three references cited below \[1-3\] will provide a deeper historical perspective. I am including a proof below that suggests to whom one could attribute this metric.

The triangle inequality of the alleged metric follows immediately from the following theorem (simplified version) of Gel'fand and Naimark \[2\] (see discussion in \[1,3\] too), so I would essentially attribute it to them. I don't know to whom can one attribute the simple results from majorization theory that are needed while applying the theorem below.

>**Theorem (GN \[2\]).**  Let $A$ and $B$ be complex matrices. Then their singular values satisfy the majorization
\begin{equation*}
\log \sigma(AB) - \log \sigma(B) \prec \log\sigma(A),
\end{equation*}
where $s(X)$ denotes the vector of singular values of an operator $X$.

Let $d(A,B)$ be defined as in the OP. Then, we immediately have the following result.

> **Theorem (Metric).** Let $A$, $B$, and $C$ be arbitrary complex matrices. Then,
\begin{equation*}
d(A,B) \le d(A,C) + d(B,C).
\end{equation*}

*Proof.* 
From Corollary GN it follows that
\begin{equation*}
\log\sigma(A^{-1}CC^{-1}B) \prec \log\sigma(A^{-1}C) + \log\sigma(C^{-1}B).
\end{equation*}
Since for $x \prec y$ it follows that $|x|\prec_w |y|$, from which it then follows that
\begin{equation*}
 |\log\sigma(A^{-1}CC^{-1}B)| \prec_w |\log\sigma(A^{-1}C)| + |\log\sigma(C^{-1}B)|.
\end{equation*}
It is easy to show that (e.g., Example II.3.13 in \[1\]) if $\Phi$ is any symmetric-gauge function, then, whenever $|x| \prec_w |y|$, we have $\Phi(x) \le \Phi(y)$. Selecting $\Phi(x) = \|x\|_p$, and applying to the above majorization we immediately obtain the desired triangle inequality.

**REFERENCES**

\[1\]. *Matrix Analysis*, R.Bhatia

\[2\]. *The relation between the unitary representations of the complex unimodular group and its unitary subgroup*, Izv. Akad. Nauk SSSR Ser. Mat. 14(1950), 239-260

\[3\]. *[The eigen- and singular values of the sum and product of linear operators][2]*, A. Markus, Uspekhi Mat. Nauk, 19:4(118) (1964), 93–123.


  [1]: http://repository.ias.ac.in/2599/1/373.pdf
  [2]: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=6224&option_lang=eng