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...]