Spectral Geometric Mean in Matrix Analysis

• Spectral Geometric Mean is a two-variable noncommutative interpolation on positive definite matrices defined via the weighted formulation A02natural_t B, capturing essential spectral properties.
• It leverages the underlying Kubo–Ando geometric mean to uniquely solve Riccati-type equations and exhibits properties such as symmetry, inverse compatibility, and unitary covariance.
• The theory extends to multivariable cases and interplays with log-majorization, norm inequalities, and operator orders, offering insights for further matrix inequality research.

Searching arXiv for recent and foundational papers on spectral geometric mean. The spectral geometric mean is a two-variable mean on the cone of positive definite matrices or operators, introduced by Fiedler–Pták and commonly written

$$A\natural_t B=(A^{-1}\sharp B)^t\,A\,(A^{-1}\sharp B)^t,\qquad A,B\in\mathbb P,\ t\in[0,1],$$

with $A\natural B:=A\natural_{1/2}B$, where

$A\sharp_t B=A^{1/2}(A^{-1/2}BA^{-1/2})^tA^{1/2}$

is the Kubo–Ando geometric mean. In the commuting case it reduces to the scalar-style interpolation $A^{1-t}B^t$, but in the noncommutative case it is a distinct object, not a Kubo–Ando mean, and it is studied through spectral identities, log-majorization, nonlinear equations, and geometric constructions rather than through monotonicity in the Löwner order (Gan et al., 2021, Kim et al., 16 May 2026).

1. Origin, notation, and basic definition

For positive definite matrices $A,B\in\mathbb P_n$, the unweighted spectral geometric mean is

$$A\natural B=(A^{-1}\sharp B)^{1/2}\,A\,(A^{-1}\sharp B)^{1/2},$$

and the weighted version is

$$A\natural_t B=(A^{-1}\sharp B)^t\,A\,(A^{-1}\sharp B)^t,\qquad t\in[0,1].$$

The weighted path satisfies $A\natural_0B=A$, $A\natural_1B=B$, and $A\natural B=A\natural_{1/2}B$. The standard geometric mean $A\natural B:=A\natural_{1/2}B$0 appears twice: first in the auxiliary factor $A\natural B:=A\natural_{1/2}B$1, and then through functional calculus applied to that factor (Gan et al., 2021, Gan et al., 2022).

A basic structural characterization is that $A\natural B:=A\natural_{1/2}B$2 is the unique positive definite solution $A\natural B:=A\natural_{1/2}B$3 of

$A\natural B:=A\natural_{1/2}B$4

This Riccati-type formulation makes the mean natural from the viewpoint of geometric-mean equations rather than Kubo–Ando theory. In later work this perspective was extended to broader two-parameter families $A\natural B:=A\natural_{1/2}B$5 that contain $A\natural B:=A\natural_{1/2}B$6 as the case $A\natural B:=A\natural_{1/2}B$7, and also to other weighted extensions reducing to the original spectral mean at $A\natural B:=A\natural_{1/2}B$8 (Gan et al., 2022, Seo et al., 28 Dec 2025).

The commuting case plays a normalizing role throughout the theory: $A\natural B:=A\natural_{1/2}B$9 Accordingly, the spectral geometric mean is a noncommutative extension of the scalar geometric interpolation $A\sharp_t B=A^{1/2}(A^{-1/2}BA^{-1/2})^tA^{1/2}$0, but one whose noncommutative behavior is governed by spectral similarity and majorization rather than order monotonicity (Kim, 2020, Gan et al., 2023).

2. Structural identities and the reason for the term “spectral”

The defining algebraic identities of the weighted spectral geometric mean include symmetry,

$A\sharp_t B=A^{1/2}(A^{-1/2}BA^{-1/2})^tA^{1/2}$1

inverse compatibility,

$A\sharp_t B=A^{1/2}(A^{-1/2}BA^{-1/2})^tA^{1/2}$2

positive homogeneity,

$A\sharp_t B=A^{1/2}(A^{-1/2}BA^{-1/2})^tA^{1/2}$3

unitary covariance,

$A\sharp_t B=A^{1/2}(A^{-1/2}BA^{-1/2})^tA^{1/2}$4

and the interpolation law

$A\sharp_t B=A^{1/2}(A^{-1/2}BA^{-1/2})^tA^{1/2}$5

In matrix form it also satisfies the determinant identity

$A\sharp_t B=A^{1/2}(A^{-1/2}BA^{-1/2})^tA^{1/2}$6

These properties make $A\sharp_t B=A^{1/2}(A^{-1/2}BA^{-1/2})^tA^{1/2}$7 a coherent interpolation path even though it is not a Kubo–Ando mean (Gan et al., 2023, Gan et al., 2023).

The term “spectral” comes from its relation to the product $A\sharp_t B=A^{1/2}(A^{-1/2}BA^{-1/2})^tA^{1/2}$8. A standard fact is that $A\sharp_t B=A^{1/2}(A^{-1/2}BA^{-1/2})^tA^{1/2}$9 is positively similar to $A^{1-t}B^t$0, and hence $A^{1-t}B^t$1 has the same spectrum as $A^{1-t}B^t$2. Equivalently, the eigenvalues of $A^{1-t}B^t$3 are the positive square roots of the eigenvalues of $A^{1-t}B^t$4. In the commuting case this collapses to the classical identity $A^{1-t}B^t$5 (Kim, 2020, Gan et al., 2021).

A useful Riccati identity makes this relation explicit. If

$A^{1-t}B^t$6

then $A^{1-t}B^t$7 is the unique positive definite solution of $A^{1-t}B^t$8, and

$A^{1-t}B^t$9

This simultaneously rewrites the spectral mean and the Bures–Wasserstein cross terms in a common factorized form, which underlies several later majorization results (Vuong et al., 22 May 2026).

3. Order-theoretic position among matrix means

The spectral geometric mean is systematically compared with the metric geometric mean, the log-Euclidean mean, and the Wasserstein mean. For all $A,B\in\mathbb P_n$0 and $A,B\in\mathbb P_n$1,

$A,B\in\mathbb P_n$2

so the metric mean is log-majorized by the spectral mean. This relation is spectral rather than Löwner-theoretic: the literature also records concrete counterexamples showing that $A,B\in\mathbb P_n$3 need not hold in general (Gan et al., 2021).

A sharper hierarchy places the spectral mean between the log-Euclidean and Wasserstein means: $A,B\in\mathbb P_n$4 Here

$A,B\in\mathbb P_n$5

is the two-variable Wasserstein mean. The comparison with $A,B\in\mathbb P_n$6 is generally only weak log-majorization; it cannot in general be upgraded to full log-majorization because of the determinant inequality for the Wasserstein mean (Gan et al., 2023).

A further refinement uses the “near order”

$A,B\in\mathbb P_n$7

This order is weaker than the Löwner order but stronger than eigenvalue-by-eigenvalue order. In this framework one has

$A,B\in\mathbb P_n$8

and equality holds if and only if $A,B\in\mathbb P_n$9. The same work places the spectral mean into a longer order chain,

$$A\natural B=(A^{-1}\sharp B)^{1/2}\,A\,(A^{-1}\sharp B)^{1/2},$$0

This identifies the spectral mean as spectrally larger than several multiplicative interpolants but still below the Wasserstein and arithmetic means in the relevant senses (Gan et al., 2023).

The same majorization viewpoint persists under powering. For $$A\natural B=(A^{-1}\sharp B)^{1/2}\,A\,(A^{-1}\sharp B)^{1/2},$$1,

$$A\natural B=(A^{-1}\sharp B)^{1/2}\,A\,(A^{-1}\sharp B)^{1/2},$$2

so the map $$A\natural B=(A^{-1}\sharp B)^{1/2}\,A\,(A^{-1}\sharp B)^{1/2},$$3 is log-majorization increasing. This is one reason the spectral mean is described as “the dominant one” among several natural interpolants (Gan et al., 2021).

4. Nonlinear equations, semi-metric geometry, and asymptotic limits

The spectral geometric mean admits a nonlinear characterization that is particularly important in recent multivariable work. For $$A\natural B=(A^{-1}\sharp B)^{1/2}\,A\,(A^{-1}\sharp B)^{1/2},$$4 and $$A\natural B=(A^{-1}\sharp B)^{1/2}\,A\,(A^{-1}\sharp B)^{1/2},$$5, the unique positive definite solution of

$$A\natural B=(A^{-1}\sharp B)^{1/2}\,A\,(A^{-1}\sharp B)^{1/2},$$6

is precisely $$A\natural B=(A^{-1}\sharp B)^{1/2}\,A\,(A^{-1}\sharp B)^{1/2},$$7. Equivalently,

$$A\natural B=(A^{-1}\sharp B)^{1/2}\,A\,(A^{-1}\sharp B)^{1/2},$$8

This equation shows that the spectral mean is singled out by a balance condition involving geometric means with the inverse variable $$A\natural B=(A^{-1}\sharp B)^{1/2}\,A\,(A^{-1}\sharp B)^{1/2},$$9 (Kim et al., 16 May 2026).

A separate geometric interpretation is based on the semi-metric

$$A\natural_t B=(A^{-1}\sharp B)^t\,A\,(A^{-1}\sharp B)^t,\qquad t\in[0,1].$$0

This $$A\natural_t B=(A^{-1}\sharp B)^t\,A\,(A^{-1}\sharp B)^t,\qquad t\in[0,1].$$1 is symmetric, nonnegative, and vanishes exactly when $$A\natural_t B=(A^{-1}\sharp B)^t\,A\,(A^{-1}\sharp B)^t,\qquad t\in[0,1].$$2, but it does not satisfy the triangle inequality in general. With respect to $$A\natural_t B=(A^{-1}\sharp B)^t\,A\,(A^{-1}\sharp B)^t,\qquad t\in[0,1].$$3, the weighted spectral mean gives a geodesic: $$A\natural_t B=(A^{-1}\sharp B)^t\,A\,(A^{-1}\sharp B)^t,\qquad t\in[0,1].$$4 and in particular

$$A\natural_t B=(A^{-1}\sharp B)^t\,A\,(A^{-1}\sharp B)^t,\qquad t\in[0,1].$$5

In this sense the unweighted spectral mean is a midpoint, contrasting with the ordinary geometric mean, which is the midpoint for the Riemannian trace metric (Gan et al., 2022, Kim, 2020).

The same papers connect $$A\natural_t B=(A^{-1}\sharp B)^t\,A\,(A^{-1}\sharp B)^t,\qquad t\in[0,1].$$6 to gyrovector geometry. In the gyrogroup structure on the cone of positive definite matrices, the ordinary weighted geometric mean appears as a gyroline, while the weighted spectral mean appears as a cogyroline. This suggests a geometric distinction between the two interpolations rather than a merely algebraic one (Kim, 2020).

An asymptotic limit links the spectral mean to Lie–Trotter interpolation: $$A\natural_t B=(A^{-1}\sharp B)^t\,A\,(A^{-1}\sharp B)^t,\qquad t\in[0,1].$$7 At the level of exponentials, one also has

$$A\natural_t B=(A^{-1}\sharp B)^t\,A\,(A^{-1}\sharp B)^t,\qquad t\in[0,1].$$8

together with monotone decrease to the limit in log-majorization: $$A\natural_t B=(A^{-1}\sharp B)^t\,A\,(A^{-1}\sharp B)^t,\qquad t\in[0,1].$$9 Thus the spectral mean is a nonlinear thickening of the log-Euclidean interpolation that collapses to it in the infinitesimal regime (Gan et al., 2022, Gan et al., 2021).

5. Multivariable spectral mean equations and the failure of global uniqueness

A recent attempt to extend the spectral geometric mean to several variables replaces the two-variable nonlinear equation by

$A\natural_0B=A$0

or equivalently

$A\natural_0B=A$1

where $A\natural_0B=A$2 is the Karcher mean. In the two-variable case, with $A\natural_0B=A$3, the unique positive definite solution is exactly $A\natural_0B=A$4 (Kim et al., 16 May 2026).

The multivariable case behaves differently. Existence holds in general: the solution set

$A\natural_0B=A$5

is nonempty for any $A\natural_0B=A$6 and $A\natural_0B=A$7. There is also a local uniqueness theorem: if all $A\natural_0B=A$8 lie in a sufficiently small neighborhood of the identity,

$A\natural_0B=A$9

then there is a unique solution in that neighborhood. However, this cannot be extended globally. An explicit $A\natural_1B=B$0 example with $A\natural_1B=B$1 shows that the same equation may have at least two distinct positive definite solutions. Accordingly, the multivariable construction yields only a set-valued mean in general, not a single-valued one (Kim et al., 16 May 2026).

Despite the loss of global uniqueness, the solution set retains many structural properties: self-duality,

$A\natural_1B=B$2

homogeneity, permutation invariance, replication invariance, unitary congruence invariance, and the determinant identity

$A\natural_1B=B$3

for every solution $A\natural_1B=B$4. Every solution also satisfies bounds between the harmonic and arithmetic means and has the same small-parameter Lie–Trotter limit as the Karcher and Wasserstein means: $A\natural_1B=B$5 This suggests that the multivariable spectral equation preserves much of the two-variable structure while sacrificing single-valuedness (Kim et al., 16 May 2026).

6. Inequalities, norm theory, and trace characterizations

A substantial inequality theory compares $A\natural_1B=B$6 with $A\natural_1B=B$7 and with scalar geometric expressions under spectral bounds. For positive invertible operators satisfying

$A\natural_1B=B$8

one obtains Hölder-type inequalities for all real orders $A\natural_1B=B$9, with constants expressed through the generalized Kantorovich constant. The same work gives operator-order comparisons between $A\natural B=A\natural_{1/2}B$0 and $A\natural B=A\natural_{1/2}B$1, norm inequalities for several real ranges of $A\natural B=A\natural_{1/2}B$2, and matrix log-majorization results such as

$A\natural B=A\natural_{1/2}B$3

It also derives applications to Tsallis relative entropies and a lower bound

$A\natural B=A\natural_{1/2}B$4

for the Umegaki relative entropy (Furuichi et al., 2024).

The Ando–Hiai problem for spectral geometric means is subtler than for Kubo–Ando means. In a two-parameter framework $A\natural B=A\natural_{1/2}B$5 containing $A\natural B=A\natural_{1/2}B$6, one has restricted Ando–Hiai implications rather than unrestricted ones. For the usual weighted spectral mean this yields

$A\natural B=A\natural_{1/2}B$7

for

$A\natural B=A\natural_{1/2}B$8

and in particular

$A\natural B=A\natural_{1/2}B$9

The corresponding matrix statement is a restricted log-majorization monotonicity for $A\natural B:=A\natural_{1/2}B$00 (Seo et al., 28 Dec 2025).

Recent work also shows that the spectral geometric mean can characterize the trace among positive linear functionals on $A\natural B:=A\natural_{1/2}B$01. A positive linear functional $A\natural B:=A\natural_{1/2}B$02 satisfies

$A\natural B:=A\natural_{1/2}B$03

if and only if $A\natural B:=A\natural_{1/2}B$04 is a nonnegative scalar multiple of the trace; the same is true for

$A\natural B:=A\natural_{1/2}B$05

The proofs use perturbations of rank-one projections corresponding to nearly parallel pure states. A related fidelity-flavored inequality is proved not to characterize the trace in the same way (Bikchentaev et al., 16 May 2026).

At the level of majorization, a sharp Heron-type comparison has been established for the spectral mean cross term: $A\natural B:=A\natural_{1/2}B$06 where

$A\natural B:=A\natural_{1/2}B$07

The coefficient $A\natural B:=A\natural_{1/2}B$08 is sharp, and at the endpoint $A\natural B:=A\natural_{1/2}B$09 weak majorization strengthens to majorization. By Ky Fan dominance this yields norm inequalities for all unitarily invariant norms (Vuong et al., 22 May 2026).

7. Distinct operator-theoretic usage involving Hadamard geometric means

A separate literature studies spectral quantities of Hadamard or Schur geometric means of positive kernel operators on Banach function spaces. There the basic object is not $A\natural B:=A\natural_{1/2}B$10 but the set-valued Hadamard weighted geometric mean

$A\natural B:=A\natural_{1/2}B$11

for bounded sets of positive kernel operators. Its generalized and joint spectral radii satisfy

$A\natural B:=A\natural_{1/2}B$12

and, in the unweighted case,

$A\natural B:=A\natural_{1/2}B$13

with parallel results for the joint spectral radius $A\natural B:=A\natural_{1/2}B$14 (Peperko, 2016).

This spectral-radius theory was later refined. For bounded sets $A\natural B:=A\natural_{1/2}B$15 of positive kernel operators, refined inequalities insert intermediate Hadamard-product expressions between the geometric mean and the ordinary product, including

$A\natural B:=A\natural_{1/2}B$16

and

$A\natural B:=A\natural_{1/2}B$17

for $A\natural B:=A\natural_{1/2}B$18, with essential spectral-radius analogues under order continuity assumptions on $A\natural B:=A\natural_{1/2}B$19 and $A\natural B:=A\natural_{1/2}B$20 (Peperko, 2018).

Further extensions establish parallel bounds for the operator norm, Hausdorff measure of noncompactness, essential spectral radius, and numerical radius of Hadamard weighted geometric means of positive kernel operators on Banach function and sequence spaces. This line of work is terminologically adjacent to the Fiedler–Pták spectral geometric mean but mathematically distinct: it concerns Hadamard-geometric constructions whose spectral growth is controlled by ordinary multiplicative products (Bogdanović et al., 2021).

The contemporary theory of the spectral geometric mean is therefore bifurcated. In finite-dimensional matrix analysis and operator theory, $A\natural B:=A\natural_{1/2}B$21 denotes the Fiedler–Pták mean, with a rich structure involving spectral similarity, nonlinear equations, majorization, and multivariable set-valued extensions. In Banach-lattice operator theory, “geometric mean” often refers instead to Hadamard constructions whose generalized and essential spectral radii satisfy refined comparison inequalities. The coexistence of these usages reflects a common emphasis on spectral control, but the underlying objects and techniques are different.