Related papers: A Note on Approximate Hadamard Matrices

It is shown that a normalized complex Hadamard matrix of order $6$ having three distinct columns, each containing at least one $-1$ entry necessarily belongs to the transposed Fourier family, or to the family of $2$-circulant complex…

The Hadamard Extension of a matrix is the matrix consisting of all Hadamard products of subsets of its rows. This construction arises in the context of identifying a mixture of product distributions on binary random variables: full column…

A characterization of $\mathbb{Z} _t \times \mathbb{Z}_2^2$-cocyclic Hadamard matrices is described, depending on the notions of {\em distributions}, {\em ingredients} and {\em recipes}. In particular, these notions lead to the…

The Hadamard maximal determinant (maxdet) problem is to find the maximum determinant D(n) of a square {+1, -1} matrix of given order n. Such a matrix with maximum determinant is called a saturated D-optimal design. We consider some cases…

Given an input matrix polynomial whose coefficients are floating point numbers, we consider the problem of finding the nearest matrix polynomial which has rank at most a specified value. This generalizes the problem of finding a nearest…

We construct orthogonal arrays OA$_{\lambda} (k,n)$ (of strength two) having a row that is repeated $m$ times, where $m$ is as large as possible. In particular, we consider OAs where the ratio $m / \lambda$ is as large as possible; these…

The rigidity of a matrix measures how many of its entries need to be changed in order to reduce its rank to some value. Good lower bounds on the rigidity of an explicit matrix would imply good lower bounds for arithmetic circuits as well as…

An analytical method for getting new complex Hadamard matrices by using mutually unbiased bases and a nonlinear doubling formula is provided. The method is illustrated with the n=4 case that leads to a rich family of eight-dimensional…

We show that a matrix is a Hermitian positive semidefinite matrix whose nonzero entries have modulus 1 if and only if it similar to a direct sum of all $1's$ matrices and a 0 matrix via a unitary monomial similarity. In particular, the only…

Let $A \in \mathbb{R}^{n \times n}$ be invertible, $x \in \mathbb{R}^n$ unknown and $b =Ax $ given. We are interested in approximate solutions: vectors $y \in \mathbb{R}^n$ such that $\|Ay - b\|$ is small. We prove that for all $0<…

We analyze the set of real and complex Hadamard matrices with additional symmetry constrains. In particular, we link the problem of existence of maximally entangled multipartite states of $2k$ subsystems with $d$ levels each to the set of…

We prove a conjecture due to Wanless about the permanent of Hadamard matrices in the particular case of Sylvester-Hadamard matrices. Namely we show that for all n greater or equal to 2, the dyadic valuation of the permanent of the…

An exchangeable random matrix is a random matrix with distribution invariant under any permutation of the entries. For such random matrices, we show, as the dimension tends to infinity, that the empirical spectral distribution tends to the…

We prove that the (non-symmetric) adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices is asymptotically almost surely invertible, assuming $\min(d,n-d)\ge C\log^2n$ for a sufficiently large constant $C>0$. The…

Let $\mathbb{P}_G([0,\infty))$ and $\mathbb{P}_G^{'}([0,\infty))$ be the sets of positive semidefinite and positive definite matrices of order $n$, respectively, with nonnegative entries, where some positions of zero entries are restricted…

The concept of matrix rigidity was introduced by Valiant(independently by Grigoriev) in the context of computing linear transformations. A matrix is rigid if it is far(in terms of Hamming distance) from any matrix of low rank. Although we…

A hollow matrix is a square matrix whose diagonal entries are all equal to zero. Define $\lambda^* = \rho^{1/2} + \rho^{-1/2} \approx 2.01980$, where $\rho$ is the unique real root of $x^3 = x + 1$. We show that for every $\lambda <…

We study the circulant complex Hadamard matrices of order $n$ whose entries are $l$-th roots of unity. For $n=l$ prime we prove that the only such matrix, up to equivalence, is the Fourier matrix, while for $n=p+q,l=pq$ with $p,q$ distinct…

We introduce qustochastic matrices as the bistochastic matrices arising from quaternionic unitary matrices by replacing each entry with the square of its norm. This is the quaternionic analogue of the unistochastic matrices studied by…

We introduce the notion of quasi-orthogonal cocycle. This is motivated in part by the maximal determinant problem for square $\{\pm 1\}$-matrices of size congruent to $2$ modulo $4$. Quasi-orthogonal cocycles are analogous to the orthogonal…