Hadamard matrix
Uncover the fascinating realm of group theory through the exploration of Hadamard matrices, jacket matrices, and cyclic groups. Delve into the concepts of equivalence, direct product, and cocycles, unveiling the beauty of mathematical structures. Discover the significance of finite groups, abelian groups, and the intricate interplay of matrices in group theory.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
Hadamard matrix F05942067
Outline Hadamard matrix Jacket matrix Group theory Cocycle & Coboundary Order 12, 16
Hadamard matrix H is a matrix of order n, n = 4, 8, 12, 16, ?? 1, 1 ???= ??? 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , 1 1 1
Equivalence ?1 ?2 ?1= ??2? for some {1, -1} permutation matrix P, Q
Jacket matrix ? = ???? ? ? 1 ? 1 ? 1= ??? 1 1 1 1 1 ? 1 1 1 1 1 1 1 ? 1 1 1 ? ? 1 1 1 1 ? ,? 1 ? = 1 ? 1 1 ? 1 1 1 1 1 ? 1 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
Jacket matrix 1 1 1 1 ? ?2 1 ? = ??2? Jacket matrix ?2 ? 3, ???? 3 = ?3= 1 1 1 1 1 1 1 1 1 1 ? ?2 1 ? ?2 1 ? ?2 1 1 1 1 ? ? ? ?2 ?2 ?2 1 ? ?2 ? ?2 1 ?2 1 ? 1 1 1 1 1 ? ?2 ?2 1 ? ? ?2 1 1 ?2 ? 1 ?2 ? 1 ?2 ? ?2 ? ? 1 ?2 ?2 ? 1 ?2 ? ?2 ? 1 ? 1 ?2 ?2 ?2 ?2 ? ? ? ?9= ?3 ?3= = ?3 ?3 ?3 ?3
Group (G, ) or (G, + ) is called a group, if ?,? ? ? ? ? ?,?,? ? ? ? ? = ? ? ? 1 ?, ?.?. ? 1 = 1 ? = ? ? ?, ? ?, ?.?. ? ? = ? ? = 1 e.g. non-zero real numbers, non-singular matrices
Group Cyclic group ?,+ ?= ? | 0 ? ? 1 , ? + ? = ? + ? ??? ? 2= 0,1 , 0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 0 Binary number with bitwise-XOR 2 2 2 2 2, direct product ?,?,? + ?,?,? = ? + ?,? + ?,? + ?
3 3 ?1= 0,0 ,?2= 0,1 ,?3= 0,2 ,?4= 1,0 ,?5= 1,1 , ?6= 1,2 ,?7= 2,0 ,?8= 2,1 ,?9= 2,2 ? = 3 3= ??,?= ???,?? ?5,6= ??5,?6= ? 1,1 , 1,2 = ?1+2= ?0= 1
Cocycle Finite group G Finitely generated abelian group C ?:? ? ? ? ?,? ? ??,? = ? ?,?? ? ?,? , ?,?,? ? ? = ? ??,??
Cocycle ?: 3 3 3 3 1,?,?2 ? ?,? = ??,? Check: ? ?,? ? ? + ?,? = ? ?,? + ? ? ?,? ??,? + ?+?,?= ??,? + ?,? + ?,?= ??,?+? + ?,?
Cocycle The set of all cocycles with ?1 ?2 ?,? = ?1?,? ?2?,? forms an abelian group ?2?,?
Coboundary ?:? ? is a function Define the coboundary ?? by ?? ?,? =? ? ? ? ? ?? Every coboundary is a cocycle ?? ?,? ?? ??,? =? ? ? ? ? ? = ?? ?,?? ?? ?,? ? ??? The set of all coboundaries is a group ?2?,?
orthogonality ??? ?,??? ? = ? ?? ?,? ? ?,? = ? ?? ?,? ? ?? 1 ?,? = ? ?? ?? 1,? ? ?? 1,? ? = ? ?? 1,? ? ?? 1,?? ? ?? 1,?? = ? ?? 1,? ? ?? 1,? ? ?? 1,? = ??? 1,??? ?? 1
Sylvester construction ?2=1 1 1 1 ?2? = ?2? 1 ?2= ?2 ?2 ?2 Cocyclic: ? 2 ? 1, 1 ?: 2 ?,? ? ?,? = 1
The Williamson construction ? ? ? ? 0 1 0 0 ? ? ? ? ? ? ? ? ? 1 0 0 0 ? ? ? 0 0 1 0 ? = , 0 0 0 , ?,?,?,? ? ?,?,?2, ,?? 1, circulant matrices ? = 1 ???= ???, ?,? ?,?,?,? ???+ ???+ ???+ ???= 4??? Cocyclic with dihedral groups
The Williamson construction 0 1 0 1 1 1 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 ? = ? + ? + ?2= , ? = 1 1 1 1 ? = ? = ? = ? ? ?2= 1 1 ???+ ???+ ???+ ???= 3? + 3? + 3?2+ 3 3? ? ?2= 12? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?12=
Dihedral group ?2? a: rotation 45 (counterclockwise) b: reflection ?2 ?3 ?4 ?5 ?6 ?7 1 ? ?? = ??7 ?? ??? = ?7 ??2 ??3 ??4 ??5 ??6 ??7 ? ?? ?2?= ?,? | ??= ?2= 1,??? = ? 1
16x16 Hadamard matrix 5 equivalent 16x16 Hadamard matrices all are cocyclic
Reference [1] Chen, Zhu, et al. "Fast cocyclic Jacket transform based on DFT." Communications, 2008. ICC'08. IEEE International Conference on. IEEE, 2008. [2] lvarez Solano, V ctor, et al. "The homological reduction method for computing cocyclic Hadamard matrices." Journal of Symbolic Computation 44.5 (2009): 558-570. [3] Horton, Jeffrey, Christos Koukouvinos, and Jennifer Seberry. "A search for Hadamard matrices constructed from Williamson matrices." (2002). [4] Flannery, D. L. "Cocyclic Hadamard matrices and Hadamard groups are equivalent." Journal of Algebra 192.2 (1997): 749-779. [5] Baliga, A., and K. J. Horadam. "Cocyclic Hadamard matrices over Zt x Z~." Australasian Journal of Combinatorics 51 (1995): 123-134.
Reference [6] lvarez, V ctor, et al. "An algorithm for computing cocyclic matrices developed over some semidirect products." International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes. Springer, Berlin, Heidelberg, 2001. [7] Horadam, Kathy J. "An introduction to cocyclic generalised Hadamard matrices." Discrete applied mathematics 102.1-2 (2000): 115-131.
