Hadamard matrix

An n×n matrix H=(hij) is a Hadamard matrixMathworldPlanetmath of order n if the entries of H are either +1 or -1 and such that HHT=nI, where HT is the transposeMathworldPlanetmath of H and I is the order n identity matrixMathworldPlanetmath.

In other words, an n×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


These matrices were first considered as Hadamard determinantsMathworldPlanetmath, because the determinant of a Hadamard matrix satisfies equality in Hadamard’s determinant theorem, which states that if X=(xij) is a matrix of order n where |xij|1 for all i and j, then


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