Graphs and Hermitian matrices: discrepancy and singular values

Graphs and Hermitian matrices: exact interlacing

We prove conditions for equality between the extreme eigenvalues of a matrix and its quotient. In particular, we give a lower bound on the largest singular value of a matrix and generalize a result of Finck and Grohmann about the largest eigenvalue of a graph.

The singularity probability of random diagonally-dominant Hermitian matrices

In this note we describe the singular locus of diagonally-dominant Hermitian matrices with nonnegative diagonal entries over the reals, the complex numbers, and the quaternions. This yields explicit expressions for the probability that such matrices, chosen at random, are singular. For instance, in the case of the identity $n \times n$ matrix perturbed by a symmetric, zero-diagonal, $\{\pm 1/(n-1)\}$-Bernoulli matrix, this probability turns out to be equal to $2^{-(n-1)(n-2)/2}$. As a corollary, we find that the probability for a $n\times n$ symmetric $\{\pm 1\}$-Bernoulli matrix to have eigen…

平凡相交子空间生成非奇异子空间的Anzahl定理 I：辛型和厄米型

Anzahl theorems for trivially intersecting subspaces generating a non-singular subspace I: symplectic and hermitian forms

本文针对辛型和厄米型非退化形式，推导出与给定子空间平凡相交且共同生成非奇异子空间的非奇异子空间数量的显式公式，从而给出精确比例并改进已知下界。

Relating multiway discrepancy and singular values of graphs and contingency tables

The $k$-way discrepancy $\disc_k (\C)$ of a rectangular array $\C$ of nonnegative entries is the minimum of the maxima of the within- and between-cluster discrepancies that can be obtained by simultaneous $k$-clusterings (proper partitions) of its rows and columns. In the main theorem, irrespective of the size of $\C$, we give the following estimate for the $k$th largest non-trivial singular value of the normalized table: $s_k \le 9\disc_{k } (\C ) (k+2 -9k\ln \disc_{k } (\C ))$, provided $\disc_{k } (\C ) <1$ and $k\le \rk (\C )$. This statement is the converse of Theorem 7 of Bolla \cite{Bol…

$\mathrm{P G}\left(s, q^2\right)$ 中 $H\left(s,q^2\right)$ 非奇异超平面的刻画

Characterization of non-singular hyperplanes of $H\left(s,q^2\right)$ in $\mathrm{P G}\left(s, q^2\right)$

本文研究射影空间中非奇异埃尔米特簇的超平面刻画问题，通过分析超平面与点及余维2子空间的交点数，建立了超平面属于埃尔米特簇的充要条件，提供了一种纯组合识别方法。

稀疏非 Hermitian 矩阵最大奇异值的大偏差

Large deviations for the largest singular value of sparse non-Hermitian matrices

本文研究常数平均度稀疏有向 Erdős–Rényi 矩阵最大奇异值的大偏差，利用团归约技术简化速率函数，证明其仅在 α=2 处转变，并推广至矩形 i.i.d. 系综。