局部半正定矩阵的特征值：非凸性与几何
Eigenvalues of locally positive semidefinite matrices: Non-convexity and Geometry
- Georgia Institute of Technology（佐治亚理工学院）
- Universität Konstanz（康斯坦茨大学）
- UiT - the Arctic University of Norway（挪威北极大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究d-局部半正定矩阵的谱几何，针对n=4、d=2的情况给出其特征值向量集的基本半代数描述，证明n≥4且d∈{2,n-2}时该集合非凸，非凸性通过复平面上的非光滑非凸极小极大点配置问题确立。
AI中文摘要：
实对称矩阵若所有d阶主子矩阵均为半正定，则称为d-局部半正定矩阵。我们研究d-局部半正定矩阵的谱几何。n阶d-局部半正定矩阵的特征值向量集已被完全理解，且当d∈{1,n-1,n}时该集合是凸的（参见文献[blekherman2022hyperbolic]）。在剩余的最小情形n=4、d=2中，文献[kozhasov2023eigenvalues]已证明特征值向量集的非凸性，但即使在该情形下其完整描述仍未知。我们通过建立4阶2-局部半正定矩阵的Fischer型不等式，给出n=4、d=2时特征值向量集的基本半代数描述，并证明当n≥4且d∈{2,n-2}时该集合非凸。非凸性通过求解复平面上的某些非光滑非凸极小极大点配置问题确立，这类问题本身可能具有研究价值。文献[nesterenko2024submatrices,sengupta2026submatrices]在矩阵分解与近似的背景下考虑了类似问题。
英文摘要：
A real symmetric matrix is called $d$-locally positive semidefinite if all of its $d \times d$ principal submatrices are positive semidefinite. We investigate the spectral geometry of $d$-locally positive semidefinite matrices. The set of vectors of eigenvalues of $d$-locally positive semidefinite matrices of size $n \times n$ is fully understood and known to be convex when $d \in \{1,n-1,n\}$ \cite{blekherman2022hyperbolic}. In the smallest remaining case $n=4, d=2$, non-convexity of the set of vectors of eigenvalues was proved in \cite{kozhasov2023eigenvalues}, but even in this case the full description was unknown. We provide a basic semialgebraic description of the set of vectors of eigenvalues for $n=4, d=2$ by establishing a Fischer-type inequality for $2$-locally positive semidefinite matrices of size $4 \times 4$ and prove non-convexity for $n \geq 4$ and $d \in \{2, n-2\}$. Non-convexity is established via solving certain non-smooth and non-convex min-max point configuration problems in the complex plane, which could be interesting in themselves. Similar problems were considered in \cite{nesterenko2024submatrices,sengupta2026submatrices} in the context of matrix decomposition and approximation.
