The Sylvester-Hadamard Matrix of Rank 2ⁿ

Abstract

We will be concerned with a particular form of the Hadamard matrix of rank 2ⁿ. This form is produced using a recursive Kronecker product. Specifically, the Hadamard matrix of interest is designated a Sylvester-Hadamard Matrix after Sylvester (1867), denoted as H _n and created by

$${H_n} = {H_1} \otimes {H_{{n - 1}}} $$

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where

$${H_1} = \left( \begin{gathered} + 1\quad + 1 \hfill \\ + 1\quad - 1 \hfill \\ \end{gathered} \right) $$

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. Thus,

$${H_2} = \left( \begin{gathered} + 1\quad + 1\quad + 1\quad + 1 \hfill \\ + 1\quad - 1\quad + 1\quad - 1 \hfill \\ + 1\quad + 1\quad - 1\quad - 1 \hfill \\ + 1\quad - 1\quad - 1\quad + 1 \hfill \\ \end{gathered} \right) $$

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and

$${H_3} = \left( \begin{gathered} + 1\quad + 1\quad + 1\quad + 1\quad + 1\quad + 1\quad + 1\quad + 1 \hfill \\ + 1\quad - 1\quad + 1\quad - 1\quad + 1\quad - 1\quad + 1\quad - 1 \hfill \\ + 1\quad + 1\quad - 1\quad - 1\quad + 1\quad + 1\quad - 1\quad - 1 \hfill \\ + 1\quad - 1\quad - 1\quad + 1\quad + 1\quad - 1\quad - 1\quad + 1 \hfill \\ + 1\quad + 1\quad + 1\quad + 1\quad - 1\quad - 1\quad - 1\quad - 1 \hfill \\ + 1\quad - 1\quad + 1\quad - 1\quad - 1\quad + 1\quad - 1\quad + 1 \hfill \\ + 1\quad + 1\quad - 1\quad - 1\quad - 1\quad - 1\quad + 1\quad + 1 \hfill \\ + 1\quad - 1\quad - 1\quad + 1\quad - 1\quad - 1\quad + 1\quad - 1 \hfill \\ \end{gathered} \right) $$

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