Let $\Bbb{S}\_{++}^n$ be the $\frac{n(n+1)}{2}$-dimensional Riemannian manifold of the symmetric positive definite (SPD) $n\times n$ real matrices.


The Log-Euclidean distance between two points of $\Bbb{S}\_{++}^n$, i.e. between two SPD matrices $A,B\in\Bbb{S}\_{++}^n$, is given by
$$
d(A,B)=\lVert\log(A)-\log(B)\rVert\_{F},
$$
where $\|\cdot\|\_{F}$ denotes the regular Frobenius norm.


We want to prove that $d\colon\Bbb{S}\_{++}^n\times\Bbb{S}\_{++}^n\to\Bbb{R}$ is a **conditionally negative definite** function, as defined below. 


For a topological space $\mathcal{X}$, the function $f\colon\mathcal{X}\times\mathcal{X}\to\Bbb{R}$ is called **conditionally negative definite** if for any $m\in\Bbb{N}$, $x\_1,\ldots,x\_m\in\mathcal{X}$, and any real numbers $c\_1,\ldots,c\_m$ for which $\sum\_{i=1}^{m}c\_i=0$, the following holds true
$$
\sum\_{i,j=1}^{m}c\_ic\_jf(x\_i,x\_j)\leq0.
$$


In our case, we want to prove that for any $m\in\Bbb{N}$, $X\_1,\ldots,X\_m\in\Bbb{S}\_{++}^n$, and any real numbers $c\_1,\ldots,c\_m$ for which $\sum\_{i=1}^{m}c\_i=0$, the following holds true
$$
\sum\_{i,j=1}^{m}c\_ic\_j\lVert\log(X\_i)-\log(X\_j)\rVert\_{F}\leq0.
$$
Thank you very much in advance!


**Edit:** In case of the squared Frobenius norm, I think it would be proven more easily, but I still need some help...



Prove that the Log-Euclidean distance is negative-definite
First, recall that if $\psi$ is negative definite, then $\exp(-\gamma \psi)$ is positive definite for all $\gamma >0$. Now, from Corollary 2.10 of [*Harmonic analysis on Semigroups* (Berg, Christensen, Ressel)](http://link.springer.com/book/10.1007%2F978-1-4612-1128-0) we also know that if $\psi$ is negative definite, then $\psi^\alpha$ is also negative definite for $0 < \alpha < 1$ as long as $\psi(x,x) \ge 0$. 


Consider now some map $f: \mathcal{X} \to \mathbb{R}^n$, and set $\psi(x,y) = \|f(x)-f(y)\|^2$. Clearly, $\psi$ is negative definite. 


In particular, choosing $\mathcal{X} = \mathbb{S}^n\_{++}$, $f \equiv \log$, and $\alpha=1/2$, from the above corollary it follows that $\|\log X - \log Y\|\_\text{F}$ is negative definite.


