On the sum of k largest singular values of graphs and matrices

Graphs and Hermitian matrices: discrepancy and singular values

We introduce a measure of discrepancy of Hermitian matrices and establish an inequality between the second singular value of a Hermitian matrix and its discrepancy. These results are applied to answer two questions of Fan Chung about graph eigenvalues.

Z形图矩阵的奇异值谱

The Spectrum of the Singular Values of Z-Shaped Graph Matrices

针对图矩阵相关研究匮乏的问题，该文确定了Z形图矩阵奇异值谱的极限分布，并将结果部分推广到$m$层Z形图矩阵，推进了对图矩阵的理解。

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…

奇异交错符号矩阵群

Groups of singular alternating sign matrices

本文研究由奇异交错符号矩阵构成的乘法群，给出构造方法，证明任意有限群均可如此表示，并探讨矩阵大小、秩与乘法阶的关系。

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。

Block LU factorization of M-matrices

It is well known that any nonsingular M-matrix admits an LU factorization into M-matrices (with L and U lower and upper triangular respectively) and any singular M-matrix is permutation similar to an M-matrix which admits an LU factorization into M-matrices. Varga and Cai establish necessary and sufficient conditions for a singular M-matrix (without permutation) to allow an LU factorization with L nonsingular. We generalize these results in two directions. First, we find necessary and sufficient conditions for the existence of an LU factorization of a singular M-matrix where L and U are both p…

Tropical bounds for eigenvalues of matrices

We show that for all k = 1,...,n the absolute value of the product of the k largest eigenvalues of an n-by-n matrix A is bounded from above by the product of the k largest tropical eigenvalues of the matrix |A| (entrywise absolute value), up to a combinatorial constant depending only on k and on the pattern of the matrix. This generalizes an inequality by Friedland (1986), corresponding to the special case k = 1.

Invertibility and Largest Eigenvalue of Symmetric Matrix Signings

The spectra of signed matrices have played a fundamental role in social sciences, graph theory, and control theory. In this work, we investigate the computational problems of identifying symmetric signings of matrices with natural spectral properties. Our results are twofold: 1. We show NP-completeness for the following three problems: verifying whether a given matrix has a symmetric signing that is positive semi-definite/singular/has bounded eigenvalues. However, we also illustrate that the complexity could substantially differ for input matrices that are adjacency matrices of graphs. 2. We e…

超临界稀疏Wigner矩阵最大特征值的大偏差

Large deviations of the largest eigenvalue of supercritical sparse Wigner matrices

针对平均度至少为维度对数阶的超临界稀疏Wigner矩阵，研究其最大特征值的大偏差上尾事件，揭示由高度顶点或大权重重团引发的机制，发现速率函数在典型值处不连续，补充了常平均度情形的已有结果。