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The cone ${\bf SPD}_n({\mathbb R})$ of symmetric positive definite matrices is endowed with a nice geometrical structure. The midpoint of the (unique) geodesic between $A$ and $B$ is the so-called geometric mean $$A\sharp B=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}.$$

What is the associated distance ?

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  • $\begingroup$ Maybe I'm being slow, but why is $A\sharp B = B\sharp A$? $\endgroup$
    – Yemon Choi
    Commented Dec 2, 2014 at 12:14
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    $\begingroup$ @Yemon Choi. I know that it is not obvious. But there are alternative characterization, where $A$ and $B$ play a symmetric role. See Exercise 198 in my list umpa.ens-lyon.fr/~serre/DPF/exobis.pdf $\endgroup$
    – Denis Serre
    Commented Dec 2, 2014 at 14:33
  • $\begingroup$ Please note that the formula noted in the answer below needs a minor fix to respect symmetry (and to ensure that $d(A,B)=\|\log \lambda(A^{-1}B)\|_2$ instead of norm of the logarithm of the singular values. $\endgroup$
    – Suvrit
    Commented Aug 13, 2017 at 18:59

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If I'm not mistaken, this midpoint comes from the structure of Riemannian symmetric space on the space of symmetric positive definite matrices. The Riemannian metric is defined by $g(S)(h_1,h_2)=trace(S^{-1}h_2S^{-1}h_2)$, where $S$ is in $SDP_n$ and $h_1,h_2$ are symmetric matrices (tangent vectors at $S$).

This metric has a lot of nice properties:

  • $Gl(n)$ acts transitively and isometrically on $SDP_n$ by $g\cdot S=gSg^{T}$, making it a homogeneous Riemannian manifold. (In particular it is complete.)

  • $S\mapsto S^{-1}$ is an isometric involution whose differential at $Id\in SDP_n$ is minus the identity. This makes $SDP_n$ a Riemannian symmetric space of non compact type (in particular it has nonpositive curvature). In the classification of Riemannian symmetric spaces it is often described as the quotient $GL(n)/O(n)$.

  • This metric also arises as the Fisher information metric on the space of centered Gaussian random vectors on $\mathbb{R}^n$ (which are characterized by their covariance matrix). This makes it useful in some applications, as a mean of treating covariance matrices in a basis independent manner.

All these properties makes it a very nice object, and it turns out that most of the objects of Riemannian geometry (connection, geodesics, curvature) can be described explicitly in terms of matrix computations. The distance is given by: $d(A,B)=\|\log(AB^{-1})\|_F$ , where $\|\ \|_F$ is the Frobenius norm. I found the formula in this article: http://arxiv.org/pdf/0807.4462v2.pdf.

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    $\begingroup$ More precisely: the correct distance is actually $d(A,B)=\|\log(B^{-1/2}AB^{-1/2})\|_F$; since $AB^{-1}$ is not symmetric one has to otherwise take a bit of care when computing $\log$ (intuitively, the distance in question is to be computed via eigenvalues not singular values). $\endgroup$
    – Suvrit
    Commented Aug 13, 2017 at 18:58

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