Spectral Decomposition and Linearization of Kubo-Ando Means

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 $σ$ with representing function $f$, we obtain an explicit decomposition of $\text{A} σ\text{B}$ in terms of the spectrum of $\text{A}<sup>{-1}\text{B}$. More precisely, we show that $\text{A} σ\text{B}$ can be expressed as a finite linear combination of matrices of the form $$\text{A}\left(\text{A}<sup>{-1}\text{B}\right)<sup>{k}$$, with coefficients depending only on $f$ and the eigenvalues of $\text{A}<sup>{-1}\text{B}$. 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.

• The paper develops a minimal spectral decomposition expressing every Kubo–Ando mean as a finite polynomial in AB⁻¹, with coefficients obtained through Lagrange interpolation of its representing function.
• For every non-affine Kubo–Ando mean, the paper proves that linearization as a combination of A and B occurs exactly when the relative spectrum of AB⁻¹ contains at most two values.
• The results provide explicit dimension-three formulas using Cardano’s method, preserve coefficients under real–complex and quaternionic embeddings, and extend polynomial decompositions to commuting alternative means.

Overview and motivation

The paper develops an explicit spectral decomposition of Kubo–Ando means on the cones of positive definite real symmetric, complex Hermitian, and quaternionic Hermitian matrices, $\mathscr{P}_{n}(\mathbb{D})$ for $\mathbb{D}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$. Recall that by the Kubo–Ando theorem, every operator mean $\sigma$ corresponds to a unique normalized operator monotone function $f$ via

$$A\,\sigma\,B = A^{\frac{1}{2}}\, f\!\left(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}\right) A^{\frac{1}{2}}.$$

Computing this expression directly requires matrix square roots, inverses, and evaluations of operator monotone functions — operations that are computationally expensive, particularly when $f$ involves logarithms. The paper's central contribution is to show that $A\,\sigma\,B$ can instead be written as a finite linear combination of powers of the single matrix $AB^{-1}$, with coefficients depending only on $f$ and the eigenvalues of $AB^{-1}$.

The two-point case and a sharp linearization criterion

The starting point is the observation that if $\mathbb{D}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$0 has spectrum of size at most two, say $\mathbb{D}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$1, then Lagrange interpolation of $\mathbb{D}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$2 at $\mathbb{D}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$3 and $\mathbb{D}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$4 by a degree-one polynomial $\mathbb{D}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$5 gives, via functional calculus, $\mathbb{D}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$6, hence

$\mathbb{D}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$7

i.e., $\mathbb{D}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$8 is linearizable — a linear combination of $\mathbb{D}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$9 and $\sigma$0 alone.

The paper then establishes a sharp converse. For a mean whose representing function $\sigma$1 is not affine, $\sigma$2 is linearizable if and only if $\sigma$3. The necessity argument uses the fact that a non-affine operator monotone function is strictly concave, so a line $\sigma$4 can agree with $\sigma$5 at no more than two spectral values. The sufficiency direction is trivial for the arithmetic mean $\sigma$6, which is always linearizable regardless of the spectrum — the criterion is therefore a characterization for non-affine means, not for all means. This dichotomy is the paper's strongest structural claim: for every non-affine Kubo–Ando mean, linearizability is governed entirely by the cardinality of the relative spectrum.

General decomposition via Lagrange interpolation

For arbitrary pairs with $\sigma$7, the paper constructs the unique degree-$\sigma$8 interpolating polynomial $\sigma$9 satisfying $f$0 on the spectrum of $f$1, and proves

$f$2

with coefficients

$f$3

where $f$4 denotes the $f$5-th elementary symmetric polynomial and $f$6 the tuple with $f$7 removed. The key algebraic step is the identity $f$8, which transfers the polynomial decomposition of $f$9 to a decomposition in powers of $$A\,\sigma\,B = A^{\frac{1}{2}}\, f\!\left(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}\right) A^{\frac{1}{2}}.$$0. A consequence is that the matrices $$A\,\sigma\,B = A^{\frac{1}{2}}\, f\!\left(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}\right) A^{\frac{1}{2}}.$$1 are linearly independent, so the decomposition is minimal — a fact used again in the compatibility analysis below.

Explicit formulas in dimension three

In $$A\,\sigma\,B = A^{\frac{1}{2}}\, f\!\left(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}\right) A^{\frac{1}{2}}.$$2 the paper eliminates the need to compute eigenvalues directly. Combining Cardano's formula with the Cayley–Hamilton theorem, the three distinct eigenvalues of $$A\,\sigma\,B = A^{\frac{1}{2}}\, f\!\left(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}\right) A^{\frac{1}{2}}.$$3 are expressed solely in terms of $$A\,\sigma\,B = A^{\frac{1}{2}}\, f\!\left(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}\right) A^{\frac{1}{2}}.$$4, $$A\,\sigma\,B = A^{\frac{1}{2}}\, f\!\left(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}\right) A^{\frac{1}{2}}.$$5, and $$A\,\sigma\,B = A^{\frac{1}{2}}\, f\!\left(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}\right) A^{\frac{1}{2}}.$$6:

• If $$A\,\sigma\,B = A^{\frac{1}{2}}\, f\!\left(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}\right) A^{\frac{1}{2}}.$$7, then $$A\,\sigma\,B = A^{\frac{1}{2}}\, f\!\left(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}\right) A^{\frac{1}{2}}.$$8 and $$A\,\sigma\,B = A^{\frac{1}{2}}\, f\!\left(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}\right) A^{\frac{1}{2}}.$$9.
• If $f$0, the two spectral values admit a closed form involving a square-root term, with a sign chosen so that $f$1.
• If $f$2, the roots are given in trigonometric form $f$3, where $f$4; the Hermitian hypothesis guarantees $f$5 and real roots.

The resulting decomposition $f$6 has coefficients $f$7 given by explicit rational expressions in $f$8 and pairwise differences of eigenvalues. The authors emphasize that this yields an effective, implementable procedure (e.g., in symbolic computation software) that avoids computing logarithms for log-based means. The same strategy extends to $f$9 via Ferrari's formula for quartics, using $A\,\sigma\,B$0, $A\,\sigma\,B$1, $A\,\sigma\,B$2, and $A\,\sigma\,B$3.

Compatibility with the real–complex embedding

The paper verifies that the decomposition is stable under the standard embedding $A\,\sigma\,B$4, $A\,\sigma\,B$5. Given the bijective correspondence between Kubo–Ando means on $A\,\sigma\,B$6 and $A\,\sigma\,B$7, the authors show that if $A\,\sigma\,B$8, then

$A\,\sigma\,B$9

with the same coefficients $AB^{-1}$0. Moreover, since each eigenvalue of $AB^{-1}$1 appears with twice its multiplicity under $AB^{-1}$2, the embedded decomposition remains minimal, so no reduction of the interpolating degree is possible. The analogous statement holds for the quaternionic embedding into $AB^{-1}$3. This invariance is a nontrivial consistency check: the decomposition coefficients are intrinsic to the mean and the relative spectrum, not artifacts of the division algebra.

Extension to alternative means

The paper also treats alternative means $AB^{-1}$4, where $AB^{-1}$5 is the geometric mean. Under the commutativity assumption $AB^{-1}$6 — which the authors state explicitly and do not remove — one has $AB^{-1}$7, and the decomposition becomes a double sum

$AB^{-1}$8

with $AB^{-1}$9 obtained from Lagrange interpolation of $f$0 at the square roots of the eigenvalues of $f$1. Because half-integer powers $f$2 reduce to polynomials in $f$3 when $f$4 (by Proposition on interpolation), every alternative mean admits a decomposition of the same polynomial form as the Kubo–Ando case.

Two further results refine this. First, for $f$5, every alternative mean is linearizable, $f$6; this recovers and extends to the entire class of alternative means the linearization phenomenon that Choi, Kim, and Lim established for the spectral geometric mean and the Wasserstein mean. Second, defining $f$7, the paper proves a sharp criterion: if $f$8 is strictly convex or strictly concave, then $f$9 is linearizable if and only if $AB^{-1}$0. This applies, for instance, to power means ($AB^{-1}$1, $AB^{-1}$2), the harmonic-type mean $AB^{-1}$3, and the quasi-Wasserstein means $AB^{-1}$4.

For non-commuting pairs, a decomposition still exists in powers of $AB^{-1}$5, but — as the authors concede — the powers $AB^{-1}$6 do not generally admit simple expressions in $AB^{-1}$7 and $AB^{-1}$8, so the commuting-case reduction to $AB^{-1}$9 is lost. This is the principal structural limitation of the alternative-means portion of the paper.

Limitations and open questions

Several restrictions are acknowledged or evident. The alternative-means decomposition requires commutativity of $\mathbb{D}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$00 and $\mathbb{D}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$01; the non-commutative analogue is only partially resolved, and the obstruction — the lack of a simple expression for powers of $\mathbb{D}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$02 — is left open. The sharp linearization criterion for Kubo–Ando means excludes affine means (the arithmetic mean is linearizable for all pairs, outside the scope of the characterization). The explicit Cardano-based formulas are worked out only for $\mathbb{D}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$03 (with a sketch for $\mathbb{D}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$04 via Ferrari), and the authors do not address the numerical stability of these closed forms near repeated eigenvalues, where the interpolation denominators $\mathbb{D}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$05 degenerate. Finally, whether an analogous minimal-decomposition theory exists for means on cones beyond $\mathbb{D}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$06 — or for the non-commutative alternative means — remains an open question raised implicitly by the paper's final remark.

Conclusion

The paper provides a complete, explicit, and minimal spectral decomposition of Kubo–Ando means as finite linear combinations of powers of $\mathbb{D}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$07, proves that linearizability of non-affine means is exactly equivalent to the relative spectrum having at most two points, delivers closed-form coefficient formulas in dimension three via Cardano and Cayley–Hamilton, establishes invariance of the decomposition under the real–complex and quaternionic embeddings, and extends the linearization phenomenon for alternative means from specific examples (the spectral geometric and Wasserstein means) to the entire class, subject to commutativity. The results reduce the computation of operator means to polynomial interpolation and eigenvalue extraction, with concrete implications for symbolic and numerical computation.