Related papers: A Note on Approximate Hadamard Matrices
The Hadamard maximal determinant problem asks for the largest n-by-n determinant with entries in {+1,-1}. When n is congruent to 1 (mod 4), the maximal excess construction of Farmakis & Kounias has been the most successful general method…
A $n\times n$ matrix $A$ has normal defect one if it is not normal, however can be embedded as a north-western block into a normal matrix of size $(n+1)\times (n+1)$. The latter is called a minimal normal completion of $A$. A construction…
We show that naturally associated to a SIC (symmetric informationally complete positive operator valued measure or SIC-POVM) in dimension d there are a number of higher dimensional structures: specifically a projector and a complex Hadamard…
For an $N \times N$ matrix $A$, its rank-$r$ rigidity, denoted $\mathcal{R}_A(r)$, is the minimum number of entries of $A$ that one must change to make its rank become at most $r$. Determining the rigidity of interesting explicit families…
In this note we present a formulae for all possible values of $(n-j)\times(n-j)$ minors of an Hadamard matrix of order $n$.
We give a precise asymptotic formula for the number of $n\times 4t$ partial Hadamard matrices in the regimes $t/n^3\to\infty$ and $t/n^3\to\Theta$ for sufficiently large fixed $\Theta$. This strengthens earlier results of de~Launey and…
M. Crouzeix formulated the following conjecture in (Integral Equations Operator Theory 48, 2004, 461--477): For every square matrix $A$ and every polynomial $p$, $$ \|p(A)\| \le 2 \max_{z\in W(A)}|p(z)|, $$ where $W(A)$ is the numerical…
The rigidity of a matrix describes the minimal number of entries one has to change to reduce matrix's rank to r. We give very simple combinatorial proof of the lower bound for the rigidity of Sylvester (special case of Hadamard) matrix that…
In this paper, we consider the equilibrium problems and also their regularized problems under the setting of Hadamard spaces. The solution to the regularized problem is represented in terms of resolvent operators. As an essential machinery…
It is shown that there exists a sequence of 3-regular graphs $\{G_n\}_{n=1}^\infty$ and a Hadamard space $X$ such that $\{G_n\}_{n=1}^\infty$ forms an expander sequence with respect to $X$, yet random regular graphs are not expanders with…
We introduce the Hadamard full propelinear codes that factorize as direct product of groups such that their associated group is $C_{2t}\times C_2$. We study the rank, the dimension of the kernel, and the structure of these codes. For…
For a given selection of rows and columns from a Fourier matrix, we give a number of tests for whether the resulting submatrix is Hadamard based on the primitive sets of those rows and columns. In particular, we demonstrate that whether a…
A real $n$-by-$n$ idempotent matrix $A$ with all entries having the same absolute value is called {\it absolutely flat}. We consider the possible ranks of such matrices and herein characterize the triples: size, constant, and rank for which…
The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. By Minc's conjecture, there exists a reachable upper bound on the permanent of 2-dimensional (0,1)-matrices. In this paper we obtain some…
Hadamard matrices, orthogonal designs and amicable orthogonal designs have a number of applications in coding theory, cryptography, wireless network communication and so on. Product designs were introduced by Robinson in order to construct…
In this paper the circulant Hadamard conjecture is proved.
A famous result of Horn and Fitzgerald is that the $\beta$-th Hadamard power of any $n\times n$ positive semi-definite (p.s.d) matrix with non-negative entries is p.s.d $\forall \beta\geq n-2$ and is not necessarliy p.s.d for $\beta< n-2,$…
We show that any two Hadamard subfactors arising from a pair of distinct complex Hadamard matrices of order 3 are either equal or conjugate by a unitary in the relative commutant of their intersection. Moreover, when the Hadamard subfactors…
We construct new symmetric Hadamard matrices of orders $92,116$, and $172$. While the existence of those of order $92$ was known since 1978, the orders $116$ and $172$ are new. Our construction is based on a recent new combinatorial array…
Although Hadamard matrices have been investigated since the nineteenth century, relatively little is known about their higher-dimensional analogues. In this paper, we introduce two constructions of Hadamard hypercubes. The first…