Matrix Means and Operator Theory

Matrix means constitute an essential toolkit in modern linear algebra and functional analysis, offering a rigorous way to interpolate and combine positive definite matrices. Rooted in the foundational Kubo–Ando theory, these means satisfy axioms of monotonicity, joint concavity and congruence invariance. Among the most prominent examples are the arithmetic, harmonic and geometric means, the latter often realised via the Ando–Li–Mathias or Karcher definitions on the Riemannian manifold of positive definite matrices. Operator theory broadens this framework by considering bounded linear operators on Hilbert spaces, enabling the extension of classical scalar inequalities—such as Jensen’s, Hölder’s and Young’s—to the operator setting. Intersection of these fields has yielded powerful methods in signal processing, machine learning, quantum information and differential geometry, where structured covariance interpolation, stability analysis and uncertainty quantification rely on well-behaved matrix averages and their spectral properties.

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Matrix Means and Operator Theory publication trend

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Technical terms

Matrix mean: A binary operation on positive definite matrices satisfying monotonicity, joint concavity and congruence invariance, generalising scalar averages.

Positive definite matrix: A Hermitian matrix whose eigenvalues are all strictly positive, forming a convex cone in matrix space.

Kubo–Ando mean: A class of operator means characterised by monotonicity under unital positive maps, including arithmetic, geometric and harmonic cases.

Loewner order: A partial order on Hermitian matrices defined by A ≤ B if B–A is positive semidefinite.

Operator convex function: A real function f for which f(λA + (1–λ)B) ≤ λf(A) + (1–λ)f(B) holds in the operator sense for all Hermitian A, B and λ∈[0,1].

Karcher mean: A multivariate geometric mean defined as the unique minimiser of the sum of squared Riemannian distances on the manifold of positive definite matrices.

References

1. Matrix-Sequences of Geometric Means in the Case of Hidden (Asymptotic) Structures. Mathematics (2025).
2. Generalized Choi–Davis–Jensen’s Operator Inequalities and Their Applications. Symmetry (2024).
3. Mixture and interpolation of the parameterized ordered means. Journal of Inequalities and Applications (2022).