Cycles and 1-unconditional matrices

Circular law for random matrices with unconditional log-concave distribution

We explore the validity of the circular law for random matrices with non i.i.d. entries. Let A be a random n \times n real matrix having as a random vector in R^{n^2} a log-concave isotropic unconditional law. In particular, the entries are uncorellated and have a symmetric law of zero mean and unit variance. This allows for some dependence and non equidistribution among the entries, while keeping the special case of i.i.d. standard Gaussian entries. Our main result states that as n goes to infinity, the empirical spectral distribution of n^{-1/2}A tends to the uniform law on the unit disc of …

Lacunary matrices

We study unconditional subsequences of the canonical basis e_rc of elementary matrices in the Schatten class S^p. They form the matrix counterpart to Rudin's Lambda(p) sets of integers in Fourier analysis. In the case of p an even integer, we find a sufficient condition in terms of trails on a bipartite graph. We also establish an optimal density condition and present a random construction of bipartite graphs. As a byproduct, we get a new proof for a theorem of Erdos on circuits in graphs.

On distance matrices of helm graphs obtained from wheel graphs with an even number of vertices

Let $n \geq 4$. The helm graph $H_n$ on $2n-1$ vertices is obtained from the wheel graph $W_n$ by adjoining a pendant edge to each vertex of the outer cycle of $W_n$. Suppose $n$ is even. Let $D := [d_{ij}]$ be the distance matrix of $H_n$. In this paper, we first show that $\det(D) = 3(n-1)2^{n-1}.$ Next, we find a matrix $Ł$ and a vector $u$ such that \[D^{-1} = -\frac{1}{2}Ł+\frac{4}{3(n-1)}uu'.\] We also prove an interlacing property between the eigenvalues of $Ł$ and $D$.

带一条加权边的循环图的拉普拉斯矩阵的特征值

Eigenvalues of the laplacian matrices of the cycles with one weighted edge

本文研究带一条加权边的循环图拉普拉斯矩阵的特征值，证明特征值仅依赖于权重实部，给出其在子区间内的定位、牛顿法收敛性及一致误差的渐近公式。