# What kind of set is this, spanned by two positive definite matrices?

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 $$B$$?