Let $A$ and $B$ be Hermitian positive definite $n\times n$ matrices over $\mathbb C$ or $\mathbb R$. Then for real $k,\ell,$ the matrix $A^kB^\ell A^k$ is well-defined and again Hermitian positive definite.
So we can iterate this and construct a set $\mathcal S$ of Hermitian p.d. matrices as follows:
- $A,B\in\mathcal S$ to start with;
- for any $U,V\in\mathcal S$ and $k\in\mathbb R$, also include $UVU$ and all matrices $U^k$ in $\mathcal S$.
Each matrix of $\mathcal S$ is thus obtained by a finite number of nested combinations of "double sided" products and powers.
Having built $\mathcal S$, we now define the span $\langle A,B\rangle$ as the subset of $\mathcal S$ of all matrices satisfying the following constraint on the "total degree" : if $A^\lambda B^\mu$ is the monomial that would result if $A,B$ were commuting, we only admit those where $\lambda ,\mu\ge0$ and $\lambda +\mu=1$. This ensures a kind of homogeneity. Note that the determinants of all those matrices are in the compact range between $\det(A)$ and $\det(B)$.
I imagine $\langle A,B\rangle$ as a compact set in $\mathbb C^{n\times n}$ or $\mathbb R^{n\times n}$ (somewhat like a sausage, or is that too intuitive?), which can be considered as a union of trajectories between $A$ and $B$, with the most prominent of them being the well-known geodesic $$\{A\sharp_t B\ | \ 0\le t\le1\} =\{A^{1/2}(A^{-1/2}BA^{-1/2})^{t}A^{1/2}\ | \ 0\le t\le1\}.$$ Note that for any $U,V\in\langle A,B\rangle$, the whole geodesic between $U$ and $V$ also belongs to $\langle A,B\rangle$. This should imply by continuity that those geodesics "fill" a whole volume in $\mathbb C^{n\times n}$ or $\mathbb R^{n\times n}$ (does it really?), thus $\langle A,B\rangle$ as a whole should have a volume.
On the other hand, if $A$ and $B$ commute, $\langle A,B\rangle$ collapses to the mere geodesic between $A$ and $B$, so the volume would be $0$.
My questions:
What can be said about $\langle A,B\rangle$?
If it is indeed a compact set, is it possible to express its "diameter in the middle" (if ever such a thing, i.e. some maximal distance, is definable) in terms of $A,B$?What about the "augmented span" obtained if we also admit finitely often additive convex combinations $tU+(1-t)V$ with $0\le t\le1$ in the construction of $\mathcal S$? Will that be strictly bigger than the convex hull of $\langle A,B\rangle$? Is it possible to express its volume or the one of the convex hull in terms of $A$ and $B$?
