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arxiv: 2606.13530 · v1 · pith:KNRKWKA5new · submitted 2026-06-11 · 🧮 math.FA

Spectral Decomposition and Linearization of Kubo-Ando Means

Raluca Dumitru , Jose Franco , Allan Merino This is my paper

Pith reviewed 2026-06-27 05:21 UTC · model grok-4.3

classification 🧮 math.FA
keywords Kubo-Ando meanspositive Hermitian matricesspectral decompositionrepresenting functionsmatrix meanslinearizationalternative means
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The pith

Kubo-Ando means of positive Hermitian matrices decompose into finite linear combinations of A times powers of A inverse B, with coefficients fixed by the representing function and the eigenvalues of A inverse B.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that any Kubo-Ando mean applied to two positive Hermitian matrices can be rewritten as a finite linear combination of terms built from powers of their scaled ratio. The coefficients in this expansion depend only on the mean's representing function and the eigenvalues of A inverse B. The authors first characterize the linear case in which every such mean admits an affine representation for suitable matrix pairs. They then give explicit coefficient formulas on the cone P_3(D) expressed through spectral invariants. The same spectral technique extends to other means and recovers explicit formulas when the matrices commute.

Core claim

Given a Kubo-Ando mean σ with representing function f, A σ B can be expressed as a finite linear combination of matrices of the form A(A^{-1}B)^k, with coefficients depending only on f and the eigenvalues of A^{-1}B. We first investigate the linear case and characterize the pairs of matrices for which every Kubo-Ando mean admits an affine representation. We then focus on the cone P_3(D), where we derive explicit formulas for the decomposition coefficients in terms of spectral invariants. Finally, we show that the same techniques extend to a broad class of alternative means, yielding explicit decompositions in the commutative setting and extending recent results of Choi, Kim, and Lim.

What carries the argument

The spectral decomposition expressing A σ B as a linear combination of A(A^{-1}B)^k whose coefficients are determined by the representing function f and the eigenvalues of A^{-1}B.

If this is right

  • Certain pairs of matrices admit an affine representation for every Kubo-Ando mean.
  • Explicit formulas for the coefficients become available on the cone P_3(D) in terms of spectral invariants.
  • The decomposition technique applies to a broad class of alternative means and produces explicit formulas in the commutative setting.
  • Recent results of Choi, Kim, and Lim on alternative means receive a direct extension via the same spectral method.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The reduction to eigenvalue data may replace iterative algorithms for computing matrix means with direct linear-algebra operations once the spectrum is known.
  • The same spectral linearization could be tested on other families of operator means that possess representing functions.
  • When the matrices commute the result collapses to a scalar identity that can be checked by direct substitution of the eigenvalues.
  • The approach supplies a concrete route for extending the decomposition beyond dimension three by tracking how the coefficients behave under block-diagonal embeddings.

Load-bearing premise

The matrices A and B are positive Hermitian over the reals, complexes or quaternions, the representing function satisfies the standard Kubo-Ando conditions, and the decomposition is valid inside the linear case or the cone P_3(D).

What would settle it

A concrete counterexample consisting of positive Hermitian matrices A and B and a valid Kubo-Ando function f for which A σ B lies outside every finite linear span of the matrices A(A^{-1}B)^k with coefficients that depend only on f and the eigenvalues of A^{-1}B.

read the original abstract

In this paper, we study the structure of Kubo-Ando means on the cone of positive Hermitian matrices over the real numbers, complex numbers, and quaternions. Given a Kubo-Ando mean $\sigma$ with representing function $f$, we obtain an explicit decomposition of $\text{A} \sigma \text{B}$ in terms of the spectrum of $\text{A}^{-1}\text{B}$. More precisely, we show that $\text{A} \sigma \text{B}$ can be expressed as a finite linear combination of matrices of the form $\text{A}\left(\text{A}^{-1}\text{B}\right)^{k}$, with coefficients depending only on $f$ and the eigenvalues of $\text{A}^{-1}\text{B}$. We first investigate the linear case and characterize the pairs of matrices for which every Kubo-Ando mean admits an affine representation. We then focus on the cone $\mathscr{P}_{3}(\mathbb{D})$, where we derive explicit formulas for the decomposition coefficients in terms of spectral invariants. Finally, we show that the same techniques extend to a broad class of alternative means, yielding explicit decompositions in the commutative setting and extending recent results of Choi, Kim, and Lim.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper establishes a spectral decomposition for Kubo-Ando means σ with representing function f on positive Hermitian matrices A, B over R, C or H: A σ B equals a finite linear combination of terms A (A^{-1}B)^k whose coefficients depend only on f and the eigenvalues of A^{-1}B. It first characterizes pairs admitting an affine representation for every such mean, then derives explicit coefficient formulas in the cone P_3(D) via spectral invariants, and finally extends the technique to a class of alternative means, recovering and generalizing results of Choi-Kim-Lim in the commutative case.

Significance. If the derivations hold, the result supplies an explicit, finite-dimensional linearization of operator means that follows directly from the spectral theorem applied to C = A^{-1/2} B A^{-1/2} together with polynomial interpolation of f at the eigenvalues of C. The explicit formulas in P_3(D) and the characterization of the linear case constitute concrete, computable expressions; the extension to alternative means adds a modest but useful generalization. These features strengthen the manuscript's contribution to the structure theory of Kubo-Ando means.

minor comments (3)
  1. [Introduction / §3] The definition of the cone P_3(D) and the precise meaning of 'spectral invariants' used for the coefficient formulas should be stated explicitly in the introduction or at the beginning of §3 rather than deferred.
  2. [Linear case section] In the linear-case characterization, the statement that the representation holds 'for every Kubo-Ando mean' would benefit from a short remark clarifying whether the affine coefficients are allowed to depend on the particular pair (A,B) or must be universal.
  3. [Final section] The extension to alternative means is stated to hold 'in the commutative setting'; a one-sentence indication of why non-commutativity obstructs the same argument would help readers assess the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the results on spectral decompositions of Kubo-Ando means, and recommendation to accept. No major comments were raised.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's derivation rests on the spectral theorem for the Hermitian positive definite operator C = A^{-1/2} B A^{-1/2} (diagonalizable over R, C, or H) together with Lagrange or Newton interpolation of the Kubo-Ando representing function f at the eigenvalues of C. This produces an explicit polynomial p such that A σ B equals A^{1/2} p(C) A^{1/2}, which by the similarity A^{-1}B = A^{-1/2} C A^{1/2} expands algebraically into a finite linear combination of terms A (A^{-1}B)^k whose coefficients depend only on f and those eigenvalues. The special cases (affine representations when ≤2 distinct eigenvalues; explicit formulas on P_3(D)) are direct restrictions of the same identity and introduce no fitted parameters or self-referential definitions. No load-bearing step reduces to a self-citation, ansatz smuggled via prior work, or renaming of an input quantity; the result is a standard consequence of the spectral theorem and polynomial interpolation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard domain assumptions from operator theory; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (2)
  • domain assumption Positive Hermitian matrices over R, C, H form a cone closed under inversion and compatible with the Kubo-Ando mean axioms.
    Standard background for defining operator means on positive definite matrices.
  • domain assumption Every Kubo-Ando mean is represented by a unique operator-monotone function f with f(1)=1.
    Core definition invoked when the paper refers to the representing function f.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On The Linearization of Alternative Means

    math.FA 2026-06 unverdicted novelty 6.0

    Proves linearization criterion for Wasserstein mean (iff commute and |Spec(A^{-1}B)|≤2) plus rigidity theorem for alternative means with f=sqrt(h) where h non-affine operator monotone.

Reference graph

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9 extracted references · 1 canonical work pages · cited by 1 Pith paper · 1 internal anchor

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