power log distance between matrices

Question

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).

Answer 1

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 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, A. Markus, Uspekhi Mat. Nauk, 19:4(118) (1964), 93–123.