Sum-product estimates for diagonal matrices

Diagonal Sums of Doubly Stochastic Matrices

Let $Ω_n$ denote the class of $n \times n$ doubly stochastic matrices (each such matrix is entrywise nonnegative and every row and column sum is 1). We study the diagonals of matrices in $Ω_n$. The main question is: which $A \in Ω_n$ are such that the diagonals in $A$ that avoid the zeros of $A$ all have the same sum of their entries. We give a characterization of such matrices, and establish several classes of patterns of such matrices.

Circulant matrices with orthogonal rows and off-diagonal entries of absolute value $1$

It is known that a real symmetric circulant matrix with diagonal entries $d\geq0$, off-diagonal entries $\pm1$ and orthogonal rows exists only of order $2d+2$ (and trivially of order $1$) [Turek and Goyeneche 2019]. In this paper we consider a complex Hermitian analogy of those matrices. That is, we study the existence and construction of Hermitian circulant matrices having orthogonal rows, diagonal entries $d\geq0$ and any complex entries of absolute value $1$ off the diagonal. As a particular case, we consider matrices whose off-diagonal entries are 4th roots of unity; we prove that the orde…

The permanent and diagonal products on the set of nonnegative matrices with bounded rank

We formulate conjectures regarding the maximum value and maximizing matrices of the permanent and of diagonal products on the set of stochastic matrices with bounded rank. We formulate equivalent conjectures on upper bounds for these functions for nonnegative matrices based on their rank, row sums and column sums. In particular we conjecture that the permanent of a singular nonnegative matrix is bounded by 1/2 times the minimum of the product of its row sums and the product of its column sums, and that the product of the elements of any diagonal of a singular nonnegative matrix is bounded by 1…

Diagonal Form of the Varchenko Matrices

Varchenko defined the Varchenko matrix associated to any real hyperplane arrangement and computed its determinant. In this paper, we show that the Varchenko matrix of a hyperplane arrangement has a diagonal form if and only if it is semigeneral, i.e., without degeneracy. In the case of semigeneral arrangement, we present an explicit computation of the diagonal form via combinatorial arguments and matrix operations, thus giving a combinatorial interpretation of the diagonal entries.

在 $\mathbb{Z}_{p^k}$ 上枚举可对角化矩阵

Enumerating Diagonalizable Matrices over $\mathbb{Z}_{p^k}$

本文研究在模 $p^k$ 整数环上可对角化方阵的计数问题，因零因子存在而比有限域情形更具挑战性，并给出枚举结果。

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…

Diagonalization of Fix-Mahonian Matrices

Consider the regular representation of the sum over all permutations weighted by the sum of their descent, inversion, and fixed point multinomials. We compute the spectrum and the multiplicities of its elements of that matrix. Note that those multinomial statistics allow to apply the result on several permutation statistics like the number of fixed points, of descents, of inversions, and the major index at the same time.

Diagonal form of the Varchenko matrices for oriented matroids

The construction of the Varchenko matrix for hyperplane arrangements, first introduced by Alexandre Varchenko, extends naturally to oriented matroids. In this paper, we generalize the theorem of Gao and Zhang by proving that the Varchenko matrix of an oriented matroid has a diagonal form if and only if the pseudohyperplane arrangement corresponding to the oriented matroid is in semigeneral position, i.e. it does not contain a degeneracy. Furthermore, we show that the Varchenko matrix of a pseudoline arrangement has a block diagonal form. This also provides an alternative combinatorial proof fo…

On the rank of Z_2-matrices with free entries on the diagonal

For an $n \times n$ matrix $M$ with entries in $\mathbb{Z}_2$ denote by $R(M)$ the minimal rank of all the matrices obtained by changing some numbers on the main diagonal of $M$. We prove that for each non-negative integer $k$ there is a polynomial in $n$ algorithm deciding whether $R(M) \leq k$ (whose complexity may depend on $k$). We also give a polynomial in $n$ algorithm computing a number $m$ such that $m/2 \leq R(M) \leq m$. These results have applications to graph drawings on non-orientable surfaces.

对角线为零的实对称矩阵与Hermitian矩阵的最小迹范数

Minimum trace norm of real symmetric and Hermitian matrices with zero diagonal

本文研究对角线为零的实对称与Hermitian矩阵的迹范数下界问题，求得迹范数与逐项L1范数比值的最小值分别为2/n和tan(π/2n)，并给出了谱范数下到最近对角矩阵距离的对偶上界。

A graphical calculus for integration over random diagonal unitary matrices

We provide a graphical calculus for computing averages of tensor network diagrams with respect to the distribution of random vectors containing independent uniform complex phases. Our method exploits the order structure of the partially ordered set of uniform block permutations. A similar calculus is developed for random vectors consisting of independent uniform signs, based on the combinatorics of the partially ordered set of even partitions. We employ our method to extend some of the results by Johnston and MacLean on the family of local diagonal unitary invariant matrices. Furthermore, our …