Hadamard functions of inverse M-Matrices

Bulk behaviour of Schur-Hadamard products of symmetric random matrices

We develop a general method for establishing the existence of the Limiting Spectral Distributions (LSD) of Schur-Hadamard products of independent symmetric patterned random matrices. We apply this method to show that the LSDs of Schur-Hadamard products of some common patterned matrices exist and identify the limits. In particular, the Schur-Hadamard product of independent Toeplitz and Hankel matrices has the semi-circular LSD. We also prove an invariance theorem that may be used to find the LSD in many examples.

Hadamard矩阵导引

Invitation to Hadamard matrices

本文介绍了实与复Hadamard矩阵的基础理论，将其视为广义Fourier矩阵，并重点探讨了复矩阵的几何与分析性质。

A Fourier-analytic Approach to Counting Partial Hadamard Matrices

In this paper, we study a family of lattice walks which are related to the Hadamard conjecture. There is a bijection between paths of these walks which originate and terminate at the origin and equivalence classes of partial Hadamard matrices. Therefore, the existence of partial Hadamard matrices can be proved by showing that there is positive probability of a random walk returning to the origin after a specified number of steps. Moreover, the number of these designs can be approximated by estimating the return probabilities. We use the inversion formula for the Fourier transform of the random…

在立方范式下计数部分Hadamard矩阵

Counting partial Hadamard matrices in the cubic regime

研究在t/n³趋近于无穷和Θ的条件下，n×4t部分Hadamard矩阵数量的渐近公式，改进了de~Launey和Levin以及Canfield之前的成果。

The Gale-Berlekamp game for complex Hadamard matrices

Associated to a complex Hadamard matrix $H\in M_N(\mathbb C)$ is the complex probability measure $μ\in\mathcal P(\mathbb C)$ describing the distribution of $φ(a,b)=<a,Hb>$, where $a,b\in\mathbb T^N$ are random. This measure is called "glow" of the matrix, due to the analogy with the Gale-Berlekamp switching game, where $H,a,b$ are real. We prove here that: (1) $μ$ becomes complex Gaussian in the $N\to\infty$ limit, (2) the universality holds as well at order 2, (3) the order 3 term seems to be quite interesting, particularly for the master Hadamard matrices, (4) in the Fourier matrix case, som…