A Note On Convexity Inequalities Of Weighted Matrix Geometric Means

…….We can consider the space of positive $n\times n$ matrix as a Reimannian metric with geodesic distance between $A,B\in\mathbf{P}_{n\times n}(\mathbb{C})$ of

with $\gamma:[a,b]\rightarrow\mathbf{P}_{n\times n}(\mathbb{C})$ being a smooth curve between $A$ and $B$ and $L(\gamma)$ its arc length.

…….This space was first introduced in Skovgaard [15] for its applications in statistics, but is of general interest as it is a quintessential example of a metric space with non-positive curvature. The corresponding unit time geodesic distance

is reached uniquely by the path

with midpoint

…….This midpoint is the ‘geometric mean’ of matrices $A$ and $B$, introduced by Pusz and Woronowicz [14] as a way of generalizing $\sqrt{xy}$ to sesquilinear forms, and the location along the geodesic for $t\in[0,1]$ is the ‘weighted geometric mean’, which has been studied in great detail with respect to its relation to the Golden-Thompson inequality, quantum entropy, relative quantum entropy, and Rényi divergences [1] [2] [6] [12] [13] .

…….Distance in this metric is invariant under conjugation, as

for any untitarily invariant norm $|||\cdot|||$, including $||\cdot||_{2}$. When $A$ and $B$ commute, we have

and

and in general,

…….However, instead of taking the Hilbert-Schmidt norm, arc length and respectively geodesic distance can be defined with the $p$-Schatten norm with $1<p<\infty$¹¹1The $p=1$ case can be defined in the same manner, but the weighted geometric mean is no longer the unique geodesic. of

now producing a Finsler manifold of non-positive curvature. Equation 1.3 is still the unique geodesic between two matrices $A$ and $B$ [4], but now with distance

Note distance is still invariant under conjugation, and a number of properties of the geodesic are preserved. Full properties of this Finsler manifold are explored in [4], both using techniques involving derivatives of the exponential map, and using log majorization.

…….Two significant advances in work have been done recently: in [9], Conde looks at the generalized Finsler manifold (ie $1<p<\infty$) and uses Clarkson–McCarthy inequalities

for $2\leq p<\infty$ and reversing for $1\leq p\leq 2$ to derive $r$-uniform convexity inequalities as defined in [3]; most significantly, Conde establishes

Then in [5], Bhatia and Holbrook look to define a geometric mean for three positive matrices and analyze this in terms of convex hulls for the Reimannian metric ($p=2$), in particular using a different method of proof relying on log majorization and conjugation invariance of the distance.

…….This paper applies an additional note to these results: using a the methodology of Bhatia and Holbrook, we can show that all inequalities must be strict unless the matrices involve $\Gamma$-commute, reducing the equality cases of these inequalities to those in $\ell^{p}$ for both two-matrix and three-matrix inequalities. This comes from a fairly simple but vital theorem: the inequality of Equation (1.8) is an equality if and only if $A$ and $B$ commute. In Section 2 we introduce the technique log majorization to prove this theorem, and in Section 3 we use it prove the equality cases of Equations (1.12)-(1.13).

…….We note that the equality case is of particular interest, because Equations (1.12)-(1.13) are used to establish coefficients such as the modulus of convexity of the space being considered. The coefficients in $\ell^{p}$ are known to be the best coefficients. Showing that inequalities are sharp except in commuting cases shows that for these matrix spaces, non-commutativity always makes things ‘worse’. This is notable, given that there are some inequalities (ie singular value rearrangement inequalities in [8]) where inequalities between positive-semidefinite matrices $A$ and $\{UBU^{\ast}\colon U\text{ unitary}\}$ are not minimized when $A$ and $UBU^{\ast}$ commute.

…….Let $\mathbf{a},\mathbf{b}\in\mathbb{R}^{n}$ with components labeled in descending order $a_{1}\geq\dots\geq a_{n}$ and $b_{1}\geq\dots\geq b_{n}$. Then $\mathbf{b}$ weakly majorizes $\mathbf{a}$, written $\mathbf{a}\prec_{w}\bf{b}$, when

and majorizes $\mathbf{a}\prec\mathbf{b}$ when the final inequality is an equality. Weak log majorization $\mathbf{a}\prec_{w(\log)}\mathbf{b}$ is similarly defined for non-negative vectors as

with log majorization $\mathbf{a}\prec_{(\log)}\mathbf{b}$ when the final inequality is an equality. Majorization applied to matrices $A$ and $B$ is understood to be applied to the vectors of their singular values respectively, and is a vital tool for proving inequalities with respect to any unitarily invariant norms.

…….This, along with some known majorization results on matrices with equality cases from [11], gives us the tools we need to prove the equality conditions:

Note that these conditions clearly imply the reduction to standard commuting when $C=I$; ie $A,B,I$ $\Gamma$-commute if and only if $[A,B]=0$.

…….We also introduce notation for the ‘exponential unit sphere’

as elements of the unit sphere appear in the traditional convexity inequalities we want to explore; we also state the inequalities in their more traditional form in $\ell^{p}$. We now have all the tools we need to address the strictness of Inequalities (1.12)-(1.13).

…….For completeness, we also add the following theorem addressing $p$-uniform convexity for $1<p\leq 2$, which does not appear in other literature: