# The Sylvester-Hadamard Matrix of Rank 2n

• John E. Hershey
Chapter
Part of the The Springer International Series in Engineering and Computer Science book series (SECS, volume 383)

## Abstract

We will be concerned with a particular form of the Hadamard matrix of rank 2 n . 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
$${H_n} = {H_1} \otimes {H_{{n - 1}}}$$
(10)
where
$${H_1} = \left( \begin{gathered} + 1\quad + 1 \hfill \\ + 1\quad - 1 \hfill \\ \end{gathered} \right)$$
(11)
. Thus,
$${H_2} = \left( \begin{gathered} + 1\quad + 1\quad + 1\quad + 1 \hfill \\ + 1\quad - 1\quad + 1\quad - 1 \hfill \\ + 1\quad + 1\quad - 1\quad - 1 \hfill \\ + 1\quad - 1\quad - 1\quad + 1 \hfill \\ \end{gathered} \right)$$
(12)
and
$${H_3} = \left( \begin{gathered} + 1\quad + 1\quad + 1\quad + 1\quad + 1\quad + 1\quad + 1\quad + 1 \hfill \\ + 1\quad - 1\quad + 1\quad - 1\quad + 1\quad - 1\quad + 1\quad - 1 \hfill \\ + 1\quad + 1\quad - 1\quad - 1\quad + 1\quad + 1\quad - 1\quad - 1 \hfill \\ + 1\quad - 1\quad - 1\quad + 1\quad + 1\quad - 1\quad - 1\quad + 1 \hfill \\ + 1\quad + 1\quad + 1\quad + 1\quad - 1\quad - 1\quad - 1\quad - 1 \hfill \\ + 1\quad - 1\quad + 1\quad - 1\quad - 1\quad + 1\quad - 1\quad + 1 \hfill \\ + 1\quad + 1\quad - 1\quad - 1\quad - 1\quad - 1\quad + 1\quad + 1 \hfill \\ + 1\quad - 1\quad - 1\quad + 1\quad - 1\quad - 1\quad + 1\quad - 1 \hfill \\ \end{gathered} \right)$$
(13)

## Authors and Affiliations

• 1
• John E. Hershey
• 2
1. 1.Oklahoma State UniversityUSA
2. 2.General Electric Corporate Research & Development CenterUSA