Correspondence of Kubo-Ando Means over Real Division Algebras and Linearization of Means

Abstract: In this paper, we establish a bijection between Kubo-Ando operator means defined on the cones $\mathscr{P}<em>{n}(\mathbb{H})$, $\mathscr{P}</em>{2n}(\mathbb{C})$, and $\mathscr{P}<em>{4n}(\mathbb{R})$. This correspondence is induced by the canonical embeddings relating quaternionic, complex, and real positive definite matrices. We investigate several structural and geometric properties preserved by these bijections, including compatibility with functional calculus, invariance under congruence transformations, and behavior with respect to natural metrics on these cones. As an application, we prove that every Kubo-Ando mean on $\mathscr{P}</em>{2}(\mathbb{D})$, where $\mathbb{D}\in\left{\mathbb{R},\mathbb{C},\mathbb{H}\right}$, admits an explicit affine expression in terms of the matrices involved. Using the embeddings above, we derive explicit formulas for operator means on special classes of real $4\times4$ positive definite matrices arising as images of the cones $\mathscr{P}<em>{2}(\mathbb{C})$ and $\mathscr{P}</em>{2}(\mathbb{H})$. In particular, we obtain trace-determinant formulas for the geometric mean in the real, complex, and quaternionic settings.

• The paper establishes bijections between Kubo–Ando means on real, complex, and quaternionic positive definite matrix cones through canonical embeddings that preserve positivity, functional calculus, Loewner order, and their shared operator-monotone representing functions.
• The embeddings scale Frobenius norms by √2 and become Log-Euclidean isometries after rescaling, ensuring that weighted Log-Euclidean barycenters commute with transfer between real, complex, and quaternionic models.
• Every Kubo–Ando mean on 2×2 matrices over ℝ, ℂ, or ℍ has an exact affine form in the input matrices, with coefficients determined by trace and determinant invariants, while the result fails for general 4×4 real matrices outside embedded complex structures.

Overview

This paper, by Jose Franco and Allan Merino (2605.27707), develops a systematic correspondence between Kubo-Ando operator means defined on the cones of positive definite matrices over the three finite-dimensional real division algebras $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$. The mechanism is the pair of canonical algebra embeddings

$$\Psi_1: M_n(\mathbb{C}) \hookrightarrow M_{2n}(\mathbb{R}), \qquad \Psi_2: M_n(\mathbb{H}) \hookrightarrow M_{2n}(\mathbb{C}),$$

which preserve adjoints, positivity, and functional calculus. The authors show that these embeddings induce bijections between Kubo-Ando means on cones of different base fields and dimensions, verify that the correspondences respect Loewner order and Log-Euclidean geometry, and then exploit them to derive explicit affine formulas for means on $2\times 2$ matrices over each division algebra, together with trace-determinant formulas for the geometric mean.

Preliminaries and the embeddings

For $$\mathbb{D} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$$, the cone $\mathscr{P}_n(\mathbb{D})$ consists of Hermitian matrices $a = a^*$ that are positive in the sense that $x^*ax > 0$ for all nonzero $x \in \mathbb{D}^n$. The exponential map is a diffeomorphism from the Hermitian space $\mathbb{C}$0 onto $\mathbb{C}$1, so real powers $\mathbb{C}$2 are well-defined via the logarithm. In the quaternionic case, eigenvalues are taken as right eigenvalues; a positive quaternionic Hermitian matrix has $\mathbb{C}$3 positive real right eigenvalues and admits a spectral decomposition $\mathbb{C}$4 with $\mathbb{C}$5 symplectic (i.e., in the appropriate compact group $\mathbb{C}$6).

The embedding $\mathbb{C}$7 extends blockwise to matrices and satisfies $\mathbb{C}$8. The key structural result identifies the image cone:

$\mathbb{C}$9

where $\mathbb{H}$0. Analogously, writing $\mathbb{H}$1, the embedding $\mathbb{H}$2 satisfies $\mathbb{H}$3 and its image is characterized by the condition $\mathbb{H}$4. This yields the chain of inclusions $\mathbb{H}$5, with images denoted $\mathbb{H}$6 and $\mathbb{H}$7.

A central technical lemma establishes compatibility of both embeddings with Borel functional calculus: for any Borel $\mathbb{H}$8,

$\mathbb{H}$9

The proof proceeds by transporting spectral decompositions through the algebra homomorphism property; this lemma underpins essentially every result that follows.

Correspondence of Kubo-Ando means

A Kubo-Ando mean on $$\Psi_1: M_n(\mathbb{C}) \hookrightarrow M_{2n}(\mathbb{R}), \qquad \Psi_2: M_n(\mathbb{H}) \hookrightarrow M_{2n}(\mathbb{C}),$$0 is a continuous binary operation satisfying the connection axioms of Kubo-Ando theory: monotonicity with respect to the Loewner order, congruence invariance $$\Psi_1: M_n(\mathbb{C}) \hookrightarrow M_{2n}(\mathbb{R}), \qquad \Psi_2: M_n(\mathbb{H}) \hookrightarrow M_{2n}(\mathbb{C}),$$1 for all $$\Psi_1: M_n(\mathbb{C}) \hookrightarrow M_{2n}(\mathbb{R}), \qquad \Psi_2: M_n(\mathbb{H}) \hookrightarrow M_{2n}(\mathbb{C}),$$2, and normalization $$\Psi_1: M_n(\mathbb{C}) \hookrightarrow M_{2n}(\mathbb{R}), \qquad \Psi_2: M_n(\mathbb{H}) \hookrightarrow M_{2n}(\mathbb{C}),$$3. The classical Kubo-Ando theorem — which the authors note holds verbatim over $$\Psi_1: M_n(\mathbb{C}) \hookrightarrow M_{2n}(\mathbb{R}), \qquad \Psi_2: M_n(\mathbb{H}) \hookrightarrow M_{2n}(\mathbb{C}),$$4 as well — gives a bijection between such means and normalized operator monotone functions $$\Psi_1: M_n(\mathbb{C}) \hookrightarrow M_{2n}(\mathbb{R}), \qquad \Psi_2: M_n(\mathbb{H}) \hookrightarrow M_{2n}(\mathbb{C}),$$5, via

$$\Psi_1: M_n(\mathbb{C}) \hookrightarrow M_{2n}(\mathbb{R}), \qquad \Psi_2: M_n(\mathbb{H}) \hookrightarrow M_{2n}(\mathbb{C}),$$6

The correspondence theorems follow directly from this representation. First, injectivity: if two means on $$\Psi_1: M_n(\mathbb{C}) \hookrightarrow M_{2n}(\mathbb{R}), \qquad \Psi_2: M_n(\mathbb{H}) \hookrightarrow M_{2n}(\mathbb{C}),$$7 agree on the subcone $$\Psi_1: M_n(\mathbb{C}) \hookrightarrow M_{2n}(\mathbb{R}), \qquad \Psi_2: M_n(\mathbb{H}) \hookrightarrow M_{2n}(\mathbb{C}),$$8, they agree everywhere, because evaluating at scalar pairs $$\Psi_1: M_n(\mathbb{C}) \hookrightarrow M_{2n}(\mathbb{R}), \qquad \Psi_2: M_n(\mathbb{H}) \hookrightarrow M_{2n}(\mathbb{C}),$$9 forces equality of the representing functions $2\times 2$0. Second, surjectivity: given a mean $2\times 2$1 on $2\times 2$2 with representing function $2\times 2$3, the same formula defines a genuine Kubo-Ando mean $2\times 2$4 on $2\times 2$5 whose restriction to the embedded cone recovers $2\times 2$6. The main results are therefore:

• Theorem (complex–real): there is a one-to-one correspondence between Kubo-Ando means on $2\times 2$7 and $2\times 2$8, characterized by $2\times 2$9.
• Theorem (quaternionic–complex): an analogous one-to-one correspondence between means on $$\mathbb{D} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$$0 and $$\mathbb{D} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$$1 via $$\mathbb{D} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$$2.

Since representing functions coincide along these chains, the correspondences compose transitively: a mean on $$\mathbb{D} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$$3 determines unique means on $$\mathbb{D} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$$4 and $$\mathbb{D} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$$5, all sharing the same operator monotone function. The authors also record a decomposition of the disjoint union of Kubo-Ando means across real matrix sizes into residue classes modulo 4, reflecting which embedded complex/quaternionic structures can live inside $$\mathbb{D} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$$6. A remark notes that the same techniques extend to indefinite $$\mathbb{D} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$$7-Hermitian settings ($$\mathbb{D} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$$8), giving correspondences $$\mathbb{D} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$$9 and their quaternionic analogues, building on prior work of Franco–Merino and Micalizzi–Tyler.

The embeddings also preserve the Loewner order, since $\mathscr{P}_n(\mathbb{D})$0 and $\mathscr{P}_n(\mathbb{D})$1 are linear and map positive semidefinite matrices to positive semidefinite matrices. An immediate consequence is that monotonicity properties of means transfer across the correspondence without loss.

Metric properties: Log-Euclidean isometries

Both embeddings scale the Frobenius norm by exactly $\mathscr{P}_n(\mathbb{D})$2:

$\mathscr{P}_n(\mathbb{D})$3

For quaternionic matrices the ordinary trace equals the reduced trace on Hermitian matrices, so no ambiguity arises in defining the norm. Combining this scaling with functional calculus applied to $\mathscr{P}_n(\mathbb{D})$4 gives $\mathscr{P}_n(\mathbb{D})$5, and hence the rescaled restrictions of the Log-Euclidean distances,

$\mathscr{P}_n(\mathbb{D})$6

make $\mathscr{P}_n(\mathbb{D})$7 and $\mathscr{P}_n(\mathbb{D})$8 isometries:

$\mathscr{P}_n(\mathbb{D})$9

This has a direct consequence for Fréchet-type statistics on these cones: weighted Log-Euclidean barycenters commute with the embeddings,

$a = a^*$0

and similarly for $a = a^*$1. Since Log-Euclidean barycenters are widely used in diffusion tensor imaging and geometric data analysis, this says that averaging data lying on an embedded complex or quaternionic structure can be performed equivalently in either model.

Linearization of means on $a = a^*$2

The second half of the paper concerns explicit formulas. The observation is elementary but consequential: by the Cayley-Hamilton theorem, the unital algebra generated by a single $a = a^*$3 matrix is at most two-dimensional, so any primary matrix function of it is an affine polynomial in the matrix itself. Concretely, if $a = a^*$4 has eigenvalues $a = a^*$5, the interpolating polynomial yields

$a = a^*$6

with the obvious limiting values when $a = a^*$7. Applying this to the Kubo-Ando formula gives the paper's linearization theorem: every Kubo-Ando mean on $a = a^*$8, for $a = a^*$9, admits the exact affine expression

$x^*ax > 0$0

where the coefficients depend only on the spectrum of $x^*ax > 0$1 and hence only on $x^*ax > 0$2 and $x^*ax > 0$3 (the Moore determinant in the quaternionic case). This is a strong claim: no approximation is involved, and it applies uniformly to all operator monotone representing functions.

As an illustration, taking $x^*ax > 0$4 recovers the Pusz-Woronowicz formula for the geometric mean of two positive $x^*ax > 0$5 matrices:

$x^*ax > 0$6

consistent with earlier work of Choi–Kim–Lim on linearity of Cartan and Wasserstein means.

Via the embeddings, the linearization transfers to real $x^*ax > 0$7 matrices in $x^*ax > 0$8. If $x^*ax > 0$9 and $x \in \mathbb{D}^n$0 has two distinct eigenvalues $x \in \mathbb{D}^n$1, each of multiplicity two, then

$x \in \mathbb{D}^n$2

with the same coefficient formulas, and the eigenvalues are recoverable from invariants of the $x \in \mathbb{D}^n$3 matrix alone:

$x \in \mathbb{D}^n$4

using $x \in \mathbb{D}^n$5 and $x \in \mathbb{D}^n$6. For the geometric mean this yields trace-determinant formulas for real $x \in \mathbb{D}^n$7 matrices in the image of $x \in \mathbb{D}^n$8, recovering the Choi–Kim–Lim result, and quaternionic analogues follow using the reduced trace and Moore determinant.

Crucially, the authors demonstrate by explicit counterexample that the affine formula fails on general elements of $x \in \mathbb{D}^n$9: taking $\mathbb{C}$00 and $\mathbb{C}$01, the geometric mean is $\mathbb{C}$02, which cannot be written as $\mathbb{C}$03 for any scalars $\mathbb{C}$04 (matching the first and last diagonal entries forces $\mathbb{C}$05, $\mathbb{C}$06, but then the second entry would be $\mathbb{C}$07). Thus the linearization phenomenon is genuinely tied to the presence of an underlying lower-dimensional division-algebra structure, not to dimension four per se.

Projection onto the embedded complex structure

The final section addresses the question of how far an arbitrary element of $\mathbb{C}$08 sits from the embedded cone $\mathbb{C}$09. Writing $\mathbb{C}$10 in block form, the authors solve the Frobenius-norm least-squares problem of finding the closest matrix commuting with $\mathbb{C}$11. Expanding the objective reduces it to minimizing $\mathbb{C}$12 over symmetric $\mathbb{C}$13, whose unique minimizer — by strict convexity of the norm — is $\mathbb{C}$14. Hence the unique closest point of $\mathbb{C}$15 to $\mathbb{C}$16 is

$\mathbb{C}$17

This is the analogue, within the positive cone, of the Fan-Hoffman theorem identifying the Hermitian part $\mathbb{C}$18 as the nearest Hermitian matrix. It provides a concrete retraction-like map from $\mathbb{C}$19 onto the embedded complex structure, complementing the correspondence theory above.

Limitations and open questions

Several boundaries of the results deserve emphasis. The linearization theorem is intrinsically two-dimensional: it relies on Cayley-Hamilton degeneracy of $\mathbb{C}$20 matrices, and the counterexample above shows it does not extend to arbitrary $\mathbb{C}$21 real matrices outside the embedded cones. Whether analogous finite expressions exist for means on $\mathbb{C}$22 with $\mathbb{C}$23, perhaps involving higher powers of $\mathbb{C}$24, is not addressed. The metric results concern the Log-Euclidean distance specifically; behavior of other natural geometries on these cones — notably the affine-invariant Riemannian trace metric and the Wasserstein/Bures metric — under $\mathbb{C}$25 and $\mathbb{C}$26 is left open, though the citation of Choi–Kim–Lim suggests such questions are active. Finally, the extension to indefinite $\mathbb{C}$27-Hermitian cones is stated in a remark without full development, and the closest-point projection is computed only for the Frobenius geometry rather than for intrinsic metrics on the cone.

Conclusion

The paper establishes that Kubo-Ando means are in canonical bijective correspondence across the real, complex, and quaternionic positive definite cones, with the correspondence implemented by the standard division-algebra embeddings and characterized entirely by the shared operator monotone representing function. The embeddings act as scaled isometries for the Log-Euclidean geometry and commute with barycenter formation, and the resulting framework yields exact affine formulas for all means on $\mathbb{C}$28 cones over each division algebra, including trace-determinant formulas for the geometric mean on embedded real $\mathbb{C}$29 matrices. The sharpness of the linearization result — valid precisely on the embedded structures and demonstrably false elsewhere — delineates clearly where division-algebra structure, rather than dimension alone, governs the form of operator means.