FoldUnfold Table of Contents Conference Matrices Conference Matrices We will now look at another type of matrix known as a conference matrix. Definition: An $n \times n$ matrix $C$ is a Conference Matrix if every entry $c_{i,j}$ is either $0$, $-1$, or $1$ and $CC^T = (n-1)I_n$. Let $C = \begin{bmatrix} 0 & 1 \\ 1 ^ 0 \end{bmatrix}$. Then $C$ is a conference matrix as every … [Read more...]

## The Kronecker Product of Two Hadamard Matrices

FoldUnfold Table of Contents The Kronecker Product of Two Hadamard Matrices The Kronecker Product of Two Hadamard Matrices Recall from The Kronecker Product of Two Matrices page that if $A$ is an $m \times n$ matrix and $B$ is an $s \times t$ matrix then the Kronecker product $A \otimes B$ of these two matrices is the $ms \times nt$ given by: (1) \begin{align} \quad A … [Read more...]

## The Kronecker Product of Two Matrices

FoldUnfold Table of Contents The Kronecker Product of Two Matrices The Kronecker Product of Two Matrices Before we look deeper into Hadamard matrices, we will need to define a special type of product between two matrices $A$ and $B$ known as their Kronecker product. Definition: Let $A$ be an $m \times n$ matrix and let $B$ be an $s \times t$ matrix. The Kronecker Product of … [Read more...]

## The Construction of Hadamard Matrices from Paley Difference Sets

FoldUnfold Table of Contents The Construction of Hadamard Matrices from Paley Difference Sets The Construction of Hadamard Matrices from Paley Difference Sets Recall from the Paley Difference Sets page that if $q = 4n - 1$ is a prime power and if $(\mathbb{Z}_q, +)$ is the additive group of integers modulo $q$ then the corresponding Paley difference set for this group is … [Read more...]

## The Existence of Hadamard Matrices

FoldUnfold Table of Contents The Existence of Hadamard Matrices The Existence of Hadamard Matrices Recall from the Hadamard Matrices page that a Hadamard matrix of order $n$ is an $n \times n$ matrix $H$ with the properties that $h_{ij} = \pm 1$ for all $i, j \in \{ 1, 2, ..., n \}$ and $HH^T = nI_n$. On the Equivalent and Normalized Hadamard Matrices page we said two … [Read more...]

## Equivalent and Normalized Hadamard Matrices

FoldUnfold Table of Contents Equivalent and Normalized Hadamard Matrices Equivalent and Normalized Hadamard Matrices Recall from the Hadamard Matrices page that a Hadamard matrix of order $n$ is an $n \times n$ matrix $H$ with the properties that $h_{ij} = \pm 1$ for all $i, j \in \{ 1, 2, ..., n \}$ and $HH^T = nI_n$. We stated two important results on the page … [Read more...]

## Hadamard Matrices

FoldUnfold Table of Contents Hadamard Matrices Hadamard Matrices Definition: A Hadamard Matrix of order $n$ is an $n \times n$ matrix $H$ with the properties that $m_{ij} = \pm 1$ for all $i, j \in \{ 1, 2, ..., n \}$ and $H H^T = nI_n$. Here the notation "$H^T$" denotes the transpose matrix of $H$ and $nI_n$ denotes the $n \times n$ matrix whose entries are all zero except … [Read more...]

## The Bruck-Ryser-Chowla Theorem

FoldUnfold Table of Contents The Bruck-Ryser-Chowla Theorem The Bruck-Ryser-Chowla Theorem So far we have looked at a wide variety of $(v, k, \lambda)$ difference sets. Of course, there exists ordered triples $(v, k, \lambda)$ which do not yield any difference sets. The famous Bruck-Ryser-Chowla theorem gives us a criterion for determining whether a set of parameters $(v, … [Read more...]

## The Development of a Difference Set

FoldUnfold Table of Contents The Development of a Difference Set The Development of a Difference Set Recall from the Difference Sets page that if $v$, $k$, and $\lambda$ are positive integers such that $2 \leq k and $(G, +)$ is a finite group of size $v$ with identity $0$, then a $(v, k, \lambda)$-difference set on $(G, +)$ is a subset $D \subseteq G$ such that $\mid D … [Read more...]

## Difference Sets in Non-Abelian Groups

FoldUnfold Table of Contents Difference Sets in Non-Abelian Groups Difference Sets in Non-Abelian Groups Recall from the Difference Sets page that if $v$, $k$, and $\lambda$ are positive integers such that $2 \leq k and $(G, +)$ is a group of order $v$ with identity element $0$, then a $(v, k, \lambda)$-difference set in this group is a nonempty proper subset $D \subset G$ … [Read more...]