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

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 containonly $+1$ entries. A Hadamard matrix in this form is said to be normalized.

Hadamard matrices are common in signal processing and coding applications.