A K3 surface associated to certain integral matrices with integral eigenvalues

Eigenvalues of Hermitian matrices with positive sum of bounded rank

We give a minimal list of inequalities characterizing the possible eigenvalues of a set of Hermitian matrices with positive semidefinite sum of bounded rank. This answers a question of A. Barvinok.

两个实对称矩阵特征值配置的条件（符号方法）

Conditions for eigenvalue configurations of two real symmetric matrices (signature approach)

本文提出了一种基于符号方法的算法，用于确定两个实对称矩阵具有特定特征值配置的条件。

Real Symmetric Matrices with Partitioned Eigenvalues

We study the real algebraic variety of real symmetric matrices with eigenvalue multiplicities determined by a partition. We present formulas for the dimension and Euclidean distance degree. We give a parametrization by rational functions. For small matrices, we provide equations; for larger matrices, we explain how to use representation theory to find equations. We describe the ring of invariants under the action of the orthogonal group. For the subvariety of diagonal matrices, we give the degree.

On eigenvalues of symmetric matrices with PSD principal submatrices

We investigate convexity properties of the set of eigenvalue tuples of $n\times n$ real symmetric matrices, whose all $k\times k$ (where $k\leq n$ is fixed) minors are positive semidefinite. It is proven that the set $λ(\mathcal{S}^{n,k})$ of eigenvalue vectors of all such matrices is star-shaped with respect to the nonnegative orthant $\mathbb{R}^n_{\geq 0}$ and not convex already when $(n,k)=(4,2)$.

两个实对称矩阵特征值配置的条件（对称多项式方法）

Conditions for eigenvalue configurations of two real symmetric matrices (symmetric polynomial approach)

本文研究了给定两个参数化的实对称矩阵和一个特征值配置，确定参数条件以使特征值具有给定配置的问题，通过将问题转化为对称多项式的实根计数问题来解决。

Hermitian matrices with a bounded number of eigenvalues

Conjugation covariants of matrices are applied to study the real algebraic variety consisting of complex Hermitian matrices with a bounded number of distinct eigenvalues. A minimal generating system of the vanishing ideal of degenerate three by three Hermitian matrices is given, and the structure of the corresponding coordinate ring as a module over the special unitary group is determined. The method applies also for degenerate real symmetric three by three matrices. For arbitrary $n$ partial information on the minimal degree component of the vanishing ideal of the variety of $n\times n$ Hermi…