Jacket matrix

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In mathematics, a Jacket matrix is a square matrix A = aij of order n whose entries are from a field (including real field, complex field, finite field ), if

AA * = A * A = nIn

where  :A * is the transpose of the matrix of inverse entries of A , i.e.

A^{\mathrm{*}}=(a_{ji}^{-1}).

Written in different form is

\forall u,v \in \{1,2,\dots,n\}, u\neq v:~\sum_i {a_{u,i} \over a_{v,i}}=0 .

[edit] Example

J_4 = \left[ \begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 1 & -2 & 2 & -1 \\ 1 & 2 & -2 & -1 \\ 1 & -1 & -1 & 1 \\ \end{array} \right], ~~~ J_4^{-1} = {1 \over 4} \left[ \begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 1 & -{1 \over 2} & {1 \over 2} & -1 \\ 1 & {1 \over 2} & -{1 \over 2} & -1 \\ 1 & -1 & -1 & 1 \\ \end{array} \right].

[edit] Jacket Matrices and other generalizations of Hadamard Matrices

Hadamard matrices are type of Jacket matrices. In the picture we present also other generalizations, and classes of matrices with similar properties.

Above are conditions, which 3 main classes of matrices must satisfy:

Orthogonal matrices:

\forall u,v \in \{1,2,\dots,n\}, u\neq v:~\sum_i {a_{u,i} . a_{v,i}}=0; ~~~ \sum_i a_{u,i}^2 = const.

Matrices with zero complex inner product:

\forall u,v \in \{1,2,\dots,n\}, u\neq v:~\sum_i {a_{u,i} \overline{a_{v,i}}}=0; ~~~ \sum_i |a_{u,i}|^2 = const.

Jacket matrices:

\forall u,v \in \{1,2,\dots,n\}, u\neq v:~\sum_i {a_{u,i} \over a_{v,i}}=0 .

[edit] References

  • M.H. Lee, The Center Weighted Hadamard Transform, IEEE Trans.1989 AS-36, (9), pp.1247-1249.
  • S.-R.Lee and M.H.Lee, On the Reverse Jacket Matrix for Weighted Hadamard Transform, IEEE Trans. on Circuit Syst.II, vol.45.no.1, pp.436-441,Mar.1998.
  • M.H. Lee, A New Reverse Jacket Transform and its Fast Algorithm, IEEE Trans. Circuits Syst.-II , vol 47, pp.39-46, 2000.
  • M.H. Lee and B.S. Rajan, A Generalized Reverse Jacket Transform, IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 48 no.7 pp 684-691, 2001.
  • J. Hou, M.H. Lee and J.Y. Park, New Polynomial Construction of Jacket Transform, IEICE Trans. Fundamentals, vol. E86-A no. 3, pp.652-659, 2003.
  • W.P. Ma and M. H. Lee, Fast Reverse Jacket Transform Algorithms, Electronics Letters, vol. 39 no. 18 , 2003
  • Moon Ho Lee, Ju Yong Park, and Jia Hou,Fast Jacket Transform Algorithm Based on Simple Matrices Factorization, IEICE Trans. Fundamental, vol.E88-A, no.8, Aug.2005.
  • Moon Ho Lee and Jia Hou, Fast Block Inverse Jacket Transform, IEEE Signal Processing Letters, vol.13. No.8, Aug.2006.
  • Jia Hou and Moon Ho Lee ,Construction of Dual OVSF Codes with Lower Correlations, IEICE Trans. Fundamentals, Vol.E89-A, No.11 pp 3363-3367, Nov 2006.
  • Jia Hou , Moon Ho Lee and Kwang Jae Lee,Doubly Stochastic Processing on Jacket Matricess, IEICE Trans. Fundamentals, vol E89-A, no.11, pp 3368-3372, Nov 2006.
  • Chang Hue Choe, M. H. Lee, Gi Yeon Hwang, Seong Hun Kim, and Hyun Seuk Yoo, Key Agreement Protocols Based on the Center Weighted Jacket Matrix as a Sysmmetric Co-cyclic Matrix, Lecture Notes in Computer Science, vol 4105, pp 121-127 Sept 2006 .
  • Ken Finlayson, Moon Ho Lee, Jennifer Seberry, and Meiko Yamada, Jacket Matrices constructed from Hadamard Matrices and Generalized Hadamard Matrices, Australasian Journal of Combinatorics, vol.35, pp 83-88, June 2006.
  • Moon Ho Lee, and Ken.Finlayson, A Simple Element Inverse Jacket Transform Coding, Information Theory Workshop 2005, ITW 2005, Proc. of IEEE ITW 2005, 28.Aug-1.Sept., New Zealand, also IEEE Signal Processing Letters, vol. 14 no. 5, May 2007.
  • Moon Ho Lee, X. D. Zhang, Fast Block Center Weighted Hadamard Transform, IEEE Trans. Circuits Syst., vol. 54 no.12 pp 2741-2745, Dec 2007.
  • Zhu Chen, Moon Ho Lee, Fast Cocyclic Jacket Transform, IEEE Signal Processing, vol. 15 no.5 May 2008.
  • Guihua Zeng, Moon Ho Lee, A Generalized Reverse Block Jacket Transform, Accepted IEEE Trans. Circuits Syst.-I, vol. 55 no.? July 2008.
  • Guihua Zeng, Yuan Li, Ying Guo and Moon Ho Lee, Stabilizer quantum codes over the Clifford algebra, J. Phys. A: Math. Theor. vol. 41, 2008.
  • M.H. Lee, N.L. Manev, Xiao-Dong Zhang, Jacket transform eigenvalue decomposition, Applied Mathematics and Computation, vol.198, 2008.