Deciding Hadamard equivalence of Hadamard matrices

Equivalence of Hadamard matrices can be decided inO(log² n) space, and hence in subexponential time. These resource bounds follow from the existence of small distinguishing sets.     

A construction for Hadamard matrices

Let 2ⁿm be the order of an Hadamard matrix. Using block Golay sequences, a class of Hadamard matrices of order (r+4ⁿ+1)4ⁿ⁺¹m² is constructed, where r is the length of a Golay sequence.     

A construction for modified generalized Hadamard matrices using QGH matrices

Let G be a group of order mu and U a normal subgroup of G of order u. Let G/U = {U ₁,U ₂, . . . ,U _m } be the set of cosets of U in G. We say a matrix H = [h _ij ] of order k with entries from G is a quasi-generalized Hadamard matrix with respect to the cosets G/U if \({\sum_{1\le t \le k} h_{it}h_{jt}^{-1} = \lambda_{ij1}U_1+\cdots+\lambda_{ijm}U_m (\exists\lambda_{ij1},\ldots, \exists \lambda_{ijm} \in \mathbb{Z})}\) for any i ≠ j. On the other hand, in our previous article we defined a modified generalized Hadamard matrix GH(s, u, λ) over a group G, from which a TD _λ (uλ, u) admitting G as a semiregular automorphism group is obtained. In this article, we present a method for combining quasi-generalized Hadamard matrices and semiregular relative difference sets to produce modified generalized Hadamard matrices.     

A spectral equivalence for Jacobi matrices

We use the classical results of Baxter and Golinskii–Ibragimov to prove a new spectral equivalence for Jacobi matrices on . In particular, we consider the class of Jacobi matrices with conditionally summable parameter sequences and find necessary and sufficient conditions on the spectral measure such that and lie in or for s1.     

A note on the Hadamard product of matrices

It is shown that the smallest eigenvalue of the Hadamard product A × B of two positive definite Hermitian matrices is bounded from below by the smallest eigenvalue of AB^T.     

An application of Sylvester matrices

In a Banach lattice, the convergence of a series of absolute values $ \sum\limits_{{k\geq 1}} {\left| {{x_k}} \right|} $ implies the unconditional convergence of the series $ \sum\limits_{{k\geq 1}} {{x_k}} $ . The converse assertion is valid only in Banach lattices order-isomorphic to M-spaces. In this paper a new proof of this fact using Sylvester series is given.     

Hadamard products of matrices

The entry-wise product of arbitrary n × ncomplex matrices is studied. The principal tools used include the Kionecker product, field of values and diagonal multiplications. Inclusion theorems for the field of values and spectrum are developed in the general case and refined in special cases. These are employed to obtain inequalities involving the real parts of the characteristic roots and the numerical radius, and previously known results are found to be special cases of several of the theorems. In addition, the case of positive stable matrices is considered and a new class of nonnegative stable matrices is introduced, studied and related to D-stability.     

Decomposition of Hadamard matrices

The decomposition is defined. The components are each as an orthogonal matrix with elements 0, ±1. In pairs they satisfy XY^T + YX^T = 0. It is conjectured that every Hadamard matrix of order mn is decomposable into m components for m = 4 or 8.     

Equivalence of Hadamard matrices

Supposem is a square-free odd integer, andA andB are any two Hadamard matrices of order 4m. We will show thatA andB are equivalent over the integers (that is,B can be obtained fromA using elementary row and column operations which involve only integers).     

On the existence of Hadamard matrices

Given any natural number q > 3 we show there exists an integer t ? [2log₂(q ? 3)] such that an Hadamard matrix exists for every order 2^sq where s > t. The Hadamard conjecture is that s = 2.This means that for each q there is a finite number of orders 2^υq for which an Hadamard matrix is not known. This is the first time such a statement could be made for arbitrary q.In particular it is already known that an Hadamard matrix exists for each 2^sq where if q = 2^m ? 1 then s ? m, if q = 2^m + 3 (a prime power) then s ? m, if q = 2^m + 1 (a prime power) then s ? m + 1.It is also shown that all orthogonal designs of types (a, b, m ? a ? b) and (a, b), 0 ? a + b ? m, exist in orders m = 2^t and 2^t+2 · 3, t ? 1 a positive integer.     

Equivalence of Butson-type Hadamard matrices

Journal of Algebraic Combinatorics - Two matrices $$H_1$$ and $$H_2$$ with entries from a multiplicative group G are said to be monomially equivalent, denoted by $$H_1cong H_2$$ , if one of the...     

M-矩阵Hadamard积的新不等式(英文)

设A和B是非奇异M-矩阵,给出了关于A和B-1的Hadamard积的最小特征值下界τ(A°B-1)的一个新估计式,该结果改进了文献[4]的结果.     

The weights of Hadamard matrices

Let $\text{Ω}{}_{\mathrm{n}}$ denote the set of all n × n Hadamard matrices. For $\text{H ∈ Ω}{}_{\mathrm{n}}$, define the weight of H to be w(H) = number of 1's in H, and $\text{w(n) =}\text{max}\text{{w(H); H ∈ Ω}{}_{\mathrm{n}}\text{}}$. In this paper, we derive upper and lower bounds for w(n).     

Mappings of Butson-type Hadamard matrices

A $\mathrm{BH}(q,n)$ Butson-type Hadamard matrix $H$ is an $n\times n$ matrix over the complex $q$th roots of unity that fulfils $H{H}^1=n{I}_n$. It is well known that a $\mathrm{BH}(4,n)$ matrix can be used to construct a $\mathrm{BH}(2,2n)$ matrix, that is, a real Hadamard matrix. This method is here generalised to construct a $\mathrm{BH}(q,pn)$ matrix from a $\mathrm{BH}(pq,n)$ matrix, where $q$ has at most two distinct prime divisors, one of them being $p$. Moreover, an algorithm for finding the domain of the mapping from its codomain in the case $p=q=2$ is developed and used to classify the $\mathrm{BH}(4,16)$ matrices from a classification of the $\mathrm{BH}(2,32)$ matrices.     

Orbit matrices of Hadamard matrices and related codes

In this paper we introduce the notion of orbit matrices of Hadamard matrices with respect to their permutation automorphism groups and show that under certain conditions these orbit matrices yield self-orthogonal codes. As a case study, we construct codes from orbit matrices of some Paley type I and Paley type II Hadamard matrices. In addition, we construct four new symmetric (100,45,20) designs which correspond to regular Hadamard matrices, and construct codes from their orbit matrices. The codes constructed include optimal, near-optimal self-orthogonal and self-dual codes, over finite fields and over ${\mathbb{Z}}_4$.     

On the asymptotic existence of Hadamard matrices

It is conjectured that Hadamard matrices exist for all orders 4t (t>0). However, despite a sustained effort over more than five decades, the strongest overall existence results are asymptotic results of the form: for all odd natural numbers k, there is a Hadamard matrix of order k²[a+blog₂k], where a and b are fixed non-negative constants. To prove the Hadamard Conjecture, it is sufficient to show that we may take a=2 and b=0. Since Seberry's ground-breaking result, which showed that we may take a=0 and b=2, there have been several improvements where b has been by stages reduced to 3/8. In this paper, we show that for all ?>0, the set of odd numbers k for which there is a Hadamard matrix of order k²2+[?log₂k] has positive density in the set of natural numbers. The proof adapts a number-theoretic argument of Erdos and Odlyzko to show that there are enough Paley Hadamard matrices to give the result.     

The excess of complex Hadamard matrices

A complex Hadamard matrix,C, of ordern has elements 1, –1,i, –i and satisfiesCC ^*=nI_nwhereC ^* denotes the conjugate transpose ofC. LetC=[c _ij] be a complex Hadamard matrix of order is called the sum ofC. (C)=|S(C)| is called the excess ofC. We study the excess of complex Hadamard matrices. As an application many real Hadamard matrices of large and maximal excess are obtained.Supported by an NSERC grant.Supported by Telecom grant 7027, an ATERB and ARC grant # A48830241.