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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)

see also:

[Examples]
  automatically generated 3/15/2005