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The Sylvester-Hadamard Matrix of Rank 2n

  • R. K. Rao Yarlagadda
  • John E. Hershey
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)

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

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

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