Definition:Hadamard Matrix

Definition

A Hadamard matrix $H$ is a square matrix such that:

Definition 1

$(1): \quad$ all the entries of $H$ are either $+1$ or $-1$

$(2): \quad$ all the rows of $H$ are pairwise orthogonal.

Definition 2

$(1): \quad$ all the entries of $H$ are either $+1$ or $-1$

$(2): \quad H H^\intercal = n \mathbf I_n$

where:

$H^\intercal$ denotes the transpose of $H$

$\mathbf I_n$ denotes the identity matrix of order $n$

given that the order of $H$ is $n$.

Examples

Arbitrary $2 \times 2$ Hadamard Matrix

An arbitrary example of a $2 \times 2$ Hadamard matrix is:

$\begin {pmatrix} 1 & 1 \\ -1 & 1 \end {pmatrix}$

Also see

• Equivalence of Definitions of Hadamard Matrix

• Results about Hadamard matrices can be found here.

Source of Name

This entry was named for Jacques Salomon Hadamard.