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Hadamard Basis


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For $ n=2^k$ there exist orthogonal bases $ u_k$ of $ \mathbb{R}^n$ with components $ u_{j,k}\in\{-1,1\}$. Combining these vectors into one matrix $ U$ we obtain

$\displaystyle U_2 =
\left(\begin{array}{rr}
1 & 1 \\ 1 & -1
\end{array}\righ...
... -1 & 1 & -1 \\
1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1
\end{array}\right)\,
.
$

In general, these matrices can be constructed by recursion

$\displaystyle U_{2\ell} =
\left(\begin{array}{cc}
U_\ell & U_\ell \\
U_\ell & -U_\ell
\end{array}\right)\,.
$

(Authors: App/Burkhardt/Höllig)

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  automatically generated 3/15/2005