Weighted Spectral Geometric Means

• Weighted spectral geometric means are defined for positive definite matrices and extend classical Kubo–Ando means with a focus on Riemannian geodesics and operator inequalities.
• They establish a log-majorization relation between the metric mean A#ₜB and the spectral mean A◊ₜB, providing refined comparisons of eigenvalue products for noncommuting matrices.
• The construction generalizes to symmetric spaces and Lie group adjoint orbits, enhancing applications in quantum information, optimization, and differential geometry.

Weighted spectral geometric means provide a critical generalization of matrix means beyond classical Kubo–Ando theory, with central relevance for matrix analysis, operator inequalities, quantum information, and the geometry of symmetric spaces. For positive definite matrices $A,B$, the principal constructions interpolated by the “metric” geometric mean $A\#_t B$ and the “spectral” geometric mean $A\natural_t B$ are linked by majorization relations, geometric properties (Riemannian geodesics), and symmetry under Lie group actions. The log-majorization ordering between these means, established with compound-matrix maps, underpins numerous applications and generalizations.

1. Definitions: Metric and Spectral Geometric Means

Let $A,B>0$ be $n\times n$ positive definite (Hermitian) matrices, and let $t\in[0,1]$.

Metric Geometric Mean (Pusz–Woronowicz):

$$A\#_t B = A^{1/2} \bigl( A^{-1/2} B A^{-1/2} \bigr)^t A^{1/2}$$

For $t=\frac{1}{2}$, this recovers the usual geometric mean $A\#B$. On the manifold of positive definite matrices, this mean admits an alternative “log-affine” representation: $$A\#_t B = \exp \bigl( (1-t) \log A + t \log B \bigr )$$ The $(A,B)$-geodesic $t \mapsto A\#_t B$ is the unique Riemannian geodesic in the affine-invariant metric.

Spectral Geometric Mean (Fiedler–Pták):

$A\natural_t B = (A^{-1}\#B)^t A (A^{-1}\#B)^t$

For $t=\frac{1}{2}$, $A\natural_{\frac{1}{2}} B$ is the original spectral mean. These means act via (operator) functional calculus, and for commuting $A,B$ reduce to $A^{1-t} B^t$.

2. Log-Majorization and Partial Product Inequalities

Log-Majorization:

Given positive vectors $x, y \in \mathbb R^n_{>0}$, $x \prec_{\log} y$ if: $$\prod_{i=1}^k x_{[i]} \leq \prod_{i=1}^k y_{[i]} \quad (k = 1,\dots,n-1), \qquad \prod_{i=1}^n x_{[i]} = \prod_{i=1}^n y_{[i]}$$ where $x_{[i]}$ is the $i^\text{th}$ largest coordinate.

Main Result:

$$\lambda(A\#_t B) \prec_{\log} \lambda(A\natural_t B)$$

where $\lambda(\cdot)$ denotes the spectrum in decreasing order. Thus, for each $k$,

$$\prod_{i=1}^k \lambda_i(A\#_t B) \leq \prod_{i=1}^k \lambda_i(A\natural_t B)$$

and $\det(A\#_t B) = \det(A\natural_t B)$.

Proof Sketch:

Via the compound-matrix mapping $C_k(\cdot)$,

$$\prod_{i=1}^k \lambda_i(A\#_t B) = \lambda_1(C_k(A)\#_t C_k(B)), \qquad \prod_{i=1}^k \lambda_i(A\natural_t B) = \lambda_1(C_k(A)\natural_t C_k(B))$$

Joint operator monotonicity (Löwner--Heinz) yields $C_k(A)\#_t C_k(B) \leq C_k(A)\natural_t C_k(B)$, confirming the majorization.

3. Extensions to Symmetric Spaces and Adjoint Orbits

Let $G$ be a real noncompact semisimple Lie group, $K$ a maximal compact subgroup, and $P=G/K$ its symmetric space. The constructions admit analogues: $$p\#_t q = p^{1/2}(p^{-1/2} q p^{-1/2})^t p^{1/2} \qquad p\natural_t q = (p^{-1}\#q)^t p (p^{-1}\#q)^t$$ These means lie in $K$-adjoint-orbit sums. There exist $u, v \in K$ with

$p\#_t q = \exp(u (\log p) + v (\log q))$

Kostant’s pre-order $\prec_G$ on $P$ (by convex hulls of Weyl group orbits) satisfies

$p\#_t q \prec_G p\natural_t q$

which, for $G=\mathrm{GL}(n,\mathbb C)$, recovers matrix majorization.

4. Spectral Properties and Special Cases

• In the commuting case ($AB=BA$), both means coincide: $A\#_t B = A\natural_t B = A^{1-t} B^t$.
• For noncommuting $A,B$, strict log-majorization holds, numerically confirmed for $2\times2$ examples.

Illustrative Example:

$$A = \begin{pmatrix} 1 & 0 \ 0 & 4 \end{pmatrix},\quad B = \begin{pmatrix} 9 & 0 \ 0 & 16 \end{pmatrix},\quad t = \frac{1}{2}$$

Gives $$A\#B = A\natural B = \begin{pmatrix} 3 & 0 \ 0 & 4 \end{pmatrix}$$.

For noncommuting matrices,

$$A = \begin{pmatrix} 6 & -3 \ -3 & 4 \end{pmatrix} ,\quad B = \begin{pmatrix} 4 & -2 \ -2 & 5 \end{pmatrix}$$

Numerical computation verifies $\lambda(A\#B) \prec_{\log} \lambda(A\natural B)$.

5. Interplay with Unitary Orbits and So’s Formula

For Hermitian $X,Y$: $e^{X/2} e^{Y} e^{X/2} = e^{UXU^* + VYV^*}$ for some unitaries $U,V$. Thus,

$$\log(e^{X/2} e^{Y} e^{X/2}) \in \mathcal O_X^U + \mathcal O_Y^V$$

where $\mathcal O_X^U$ is the unitary orbit of $X$ under $U$.

This result generalizes to adjoint orbits for noncompact semisimple Lie groups, and the matrix case is a special case of a symmetric space geodesic sum.

6. Applications and Significance

Weighted spectral means $\natural_t$ and metric geometric means $\#_t$ underlie matrix analysis inequalities, especially those involving majorization and norm estimates. They enable comparison of operator functions, refinement of Golden–Thompson-type inequalities, and the analysis of quantum relative entropy. Generalization to symmetric spaces connects this theory to differential geometry and Lie group analysis, facilitating applications ranging from quantum information to optimization on Riemannian manifolds.

Conclusion: The weighted spectral geometric mean refines the metric mean via log-majorization of eigenvalues; both admit natural symmetric space and adjoint-orbit generalizations, and form a pivotal toolkit for matrix analysis and its quantum and geometric applications (Gan et al., 2021).