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 $HH^{T}=nI,$ 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

\\begin{bmatrix}1&1\\\\1&-1\\end{bmatrix},\\begin{bmatrix}-1&1&1&1\\\\1&-1&1&1\\\\1&1&-1&1\\\\1&1&1&-1\\end{bmatrix},\\begin{bmatrix}1&1&1&1\\\\1&-1&1&-1\\\\1&1&-1&-1\\\\1&-1&-1&1\\end{bmatrix}

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 $|x_{ij}|\\leq 1$ for all $i$ and $j,$ then

det(X)\\leq 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 containonly $+1$ entries. A Hadamard matrix in this form is said to be normalized.

Hadamard matrices are common in signal processing and coding applications.