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Mathematics-Online course: Linear Algebra - Matrices - Special Matrices

Hadamard Matrices

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For $ n=2^k$ there exist orthogonal matrices $ A_k$ of dimension $ n\times n$ with entries in $ \{-1,1\}$. These are the so called Hadamard matrices.

The first Hadamard matrices are

$\displaystyle A_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 the following recursion:

$\displaystyle A_{2k} = \left(\begin{array}{cc} A_k & A_k \\ A_k & -A_k \end{array}\right)\,. $

  automatically generated 4/21/2005