AI 中文总结
研究仿射不变几何中克罗内克正定矩阵的结构化预处理，确定最近结构化逼近与最佳结构化预处理关系，表明特定条件下唯一投影是黑塞相对条件数极小值，给出相关残差、边界等结果及验证方法。
AI 中文摘要
最近的结构化逼近和最佳结构化预处理解决不同的矩阵优化问题。我们确定了仿射不变黎曼度量下克罗内克正定矩阵的精确关系。克罗内克族是封闭且测地凸的，所以每个满矩阵都有唯一的仿射不变投影。其对数残差满足偏迹正规方程，并为阿米尔乔投影求解器产生认证点和目标误差。核心结果表明，当且仅当极端谱状态允许相同的张量边缘时，这个唯一投影也是黑塞相对条件数的极小值。可计算的边缘不匹配残差在条件最优投影处消失或产生严格下降方向。两个相对谱水平总是强制投影最优性；更强的是，每个2×2克罗内克投影都是条件最优的。因此，一个显式的2×3构造是维度最小的严格分离。残差校准边界进一步界定了最佳可达到的克罗内克条件数和投影的次优性。支持性结果将经典对角和块洛纳三明治、固定基原对偶障碍和一般对数谱目标置于相同的证书语言中。给定经过验证的数值包络和向外舍入比较，区间安全推论保留了完整克罗内克测试的健全性。确定性小矩阵检查，包括独立于偏迹求解器的多启动通用对数因子预言机，验证了所述恒等式和边界。
英文摘要
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.
Comments43 pages, 1 figure; includes appendices and deterministic numerical verification. arXiv admin note: substantial text overlap with arXiv:2607.07204
