# Hadamard matrix

An $n\times n$ matrix $H=({h}_{ij})$ is a *Hadamard matrix ^{}* of order $n$ if the entries of $H$ are either
$+1$ or $-1$ and such that $H{H}^{T}=nI,$ where ${H}^{T}$ is the transpose

^{}of $H$ and $I$ is the order $n$ identity matrix

^{}.

In other words, an $n\times n$ matrix with only $+1$ and $-1$ as its elements is Hadamard if the inner product of two distinct rows is $0$ and the inner product of a row with itself is $n$.

A few examples of Hadamard matrices are

$$\left[\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill -1\hfill \end{array}\right],\left[\begin{array}{cccc}\hfill -1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill -1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill -1\hfill \end{array}\right],\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill -1\hfill & \hfill 1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill -1\hfill & \hfill -1\hfill & \hfill 1\hfill \end{array}\right]$$ |

These matrices were first considered as Hadamard determinants^{}, because the determinant of a Hadamard matrix satisfies equality in Hadamard’s determinant theorem, which states that if $X=({x}_{ij})$ is a matrix of order $n$ where $|{x}_{ij}|\le 1$ for all $i$ and $j,$ then

$$det(X)\le {n}^{n/2}$$ |

Property 1:

The order of a Hadamard matrix is $1,2$ or $4n,$ where $n$ is an integer.

Property 2:

If the rows and columns of a Hadamard matrix are permuted, the matrix remains Hadamard.

Property 3:

If any row or column is multiplied by $-1,$ the Hadamard property is retained.

Hence it is always possible to arrange to have the first row and first column of a Hadamard matrix contain
only $+1$ entries. A Hadamard matrix in this form is said to be *normalized*.

Hadamard matrices are common in signal processing and coding applications.

Title | Hadamard matrix |
---|---|

Canonical name | HadamardMatrix |

Date of creation | 2013-03-22 13:09:45 |

Last modified on | 2013-03-22 13:09:45 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 13 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 15-00 |

Classification | msc 05B20 |

Synonym | Hadamard |

Related topic | HadamardConjecture |