15-00-HadamardMatrix.tex

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\pmtitle{Hadamard matrix}
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An $n\times n$ matrix $H = (h_{ij})$ is a \emph{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
$$det(X) \leq n^{n/2}$$

\textbf{Property 1:}

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

\textbf{Property 2:}

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

\textbf{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 \emph{normalized}.

Hadamard matrices are common in signal processing and coding applications.
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