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

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

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.