Structured Preconditioning in Affine-Invariant Geometry: Projection, Certificates, and Kronecker Separation

Nearest structured approximation and best structured preconditioning solve different matrix optimization problems.

We determine their exact relation for Kronecker positive-definite matrices under the affine-invariant Riemannian metric.

The Kronecker family is closed and geodesically convex, so every full matrix has a unique affine-invariant projection.

Its logarithmic residual satisfies partial-trace normal equations and yields certified point and objective errors for an Armijo projection solver.

Our central result shows that this unique projection is also a minimizer of the Hessian-relative condition number if and only if the extreme spectral states admit identical tensor marginals.

A computable marginal-mismatch residual either vanishes at a condition-optimal projection or produces a strict descent direction.

Two relative spectral levels always force projection optimality; more strongly, every $2\times 2$ Kronecker projection is condition-optimal.

An explicit $2\times 3$ construction is therefore a dimension-minimal strict separation.

Residual-calibrated bounds further bracket the best attainable Kronecker condition number and the suboptimality of the projection.

Supporting results place classical diagonal and block Loewner sandwiches, fixed-basis primal--dual obstructions, and general log-spectral targets in the same certificate language.

Given validated numerical enclosures and outward-rounded comparisons, an interval-safe corollary preserves the soundness of the full Kronecker tests.

Deterministic small-matrix checks, including a multistart generic log-factor oracle independent of the partial-trace solver, verify the stated identities and bounds.