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| Title of object: |
Hadamard matrix |
| Canonical Name: |
HadamardMatrix |
| Type: |
Definition |
| Created on: |
2002-11-18 02:49:38.777147-05 |
| Modified on: |
2002-11-18 02:49:38.777147-05 |
| Classification: |
msc:15-00 |
| Synonyms: |
Hadamard matrix=Hadamard |
Preamble:
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Content:
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An $n\times n$ matrix $H = h_{ij}$ is an 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 an 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 $i,j$ then
$$det(X) \leq n^{n/2}$$
\textbf{property 1:}
The order of an Hadamard matrix is $1, 2$ or $4n,$ where $n$ is an integer.
\textbf{property 2:}
If the rows and columns of an 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 an Hadamard matrix contain
only $+1$ entries. An Hadamard matrix in this form is said to be \emph{normalized}.
Hadamard matrices are common in signal proecessing and coding applications. |
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