Bures Geodesics and Restricted Barycenters for Kronecker Positive Definite Matrices

TL;DR

This paper explores the geometry of Kronecker positive definite matrices under Bures--Wasserstein metrics, identifying conditions for geodesic membership and providing formulas for restricted barycenters.

Contribution

It characterizes when Bures geodesics stay within the Kronecker model and derives exact formulas for specific restricted barycenter problems.

Findings

Geodesics stay in the model only in one-factor cases.

Membership criterion involves a partial-trace residual.

Exact formulas for restricted barycenters in specific cases.

Abstract

We study the extrinsic Bures--Wasserstein geometry of the determinant-normalized Kronecker model $\mcK_n=\{V\ot U:U,V\in\Sp^n,\ \det U=1\}\subset\Sp^{n^2}$, asking when the ambient Bures geodesic between two Kronecker positive definite matrices can remain in this lower-dimensional model. Local membership near an endpoint is shown to be equivalent to membership of the whole segment, and this happens exactly in the one-factor cases: either $U_1=U_0$ or $V_1$ is a positive scalar multiple of $V_0$. Consequently, any endpoint pair not confined to these one-factor alternatives leaves the model immediately. The criterion is expressed by a partial-trace residual. In fixed commuting charts it becomes an equivalent rank-one square-root profile and yields computable departure diagnostics. We also obtain exact formulas for two restricted barycenter problems: fixed commuting-coordinate slices, solved by Perron singular vectors, and one-factor subfamilies, reduced to standard Bures--Wasserstein barycenters on $\Sp^n$.