Why Jacket Matrices? 1 1 T [][]AN a ij Moon Ho Lee (E-mail:[email protected]) wcu.chonbuk.ac.kr, mdmc.chonbuk.ac.kr Institute of Information & Communication Chonbuk National University Jeonju, 561-756, Korea Tel: +82632702463 Fax: +82632704166 htttp://en.wikipedia.org/wiki/Category:Matrices htttp://en.wikipedia.org/wiki/Jacket:Matrix 1 http://en.wikipedia.org/wiki/user:leejacket 2 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + + -j -j j j - - + + - - + + - - + + j j -j -j - - + + - - + + - - + - -j j j -j - + + - - + + - - + + - j -j -j j - + + - - + + - - + + + j j -j -j - - + + + + - - - - + + -j -j j j - - + + + + - - - - + - j -j -j j - + + - + - - + - + + - -j j j -j - + + - + - - + - + + + - - - - + + + + - - - - + + + + - - - - + + + + - - - - + + + - - + - + + - + - - + - + + - + - - + - + + - + - - + - + + - Real Domain Complex Domain The basic idea was motivated by the cloths of Jacket. As our two sided Jacket is inside and outside compatible, at least two positions of a Jacket matrix are replaced by their inverse; these elements are changed in their position and are moved, for example, from inside of the middle circle to outside or from to inside without loss of sign. 3 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. Written in different form is a u, v {1,2,...,n }, u v : u, i 0 av, i The inverse form which is only from the entrywise inverse and transpose : Jacket Matrices T j0,0 j0,1 .... j0,n1 1/ j0,0 1/ j0,1 .... 1/ j0,n1 j j .... j 1/ j 1/ j .... 1/ j 1,0 1,1 1,n1 1 1 1,0 1,1 1,n1 J Jmn mn C 1/ jm1,0 1/ jm1,1 .... 1/ jm1,n1 jm1,0 jm1,1 .... jm1,n1 Orthogonal: u , v {1,2,..., n }, u v : a . a 0; a2 const . u,,, i v i u i 4 i i 5 6 Category : Matrices (from Http://en.wikipedia.org/wiki/Category:Matrices) H I J L ▪Hadamard matrix ▪Identity matrix ▪Jacket matrix ▪Laplacian matrix ▪Hamiltonian matrix ▪Incidence matrix ▪Jacobian matrix ▪Lehmer matrix ▪Hankel matrix ▪Integer matrix and determinant ▪Leslie matrix ▪Hasse-Witt matrix ▪Invertible matrix ▪Jones calculus ▪Levinson recursion ▪Hat matrix ▪Involutory matrix K ▪List of matrices ▪Hermitian matrix ▪Irregular matrix ▪Kernel (matrix) ▪Hessenberg matrix ▪Krawtchouk matrices 7 Jacket Basic Concept from Center Weighted Hadamard 1 1 1 1 1 1 1 1 1 2 2 1 1 1/ 2 1/ 2 1 WH ,WH 1 4 1 2 2 1 4 1 1/ 2 1/ 2 1 1 1 1 1 1 1 1 1 1 1 WH N WH N / 2 H 2 where H 2 1 1 Sparse matrix and its relation to construction of center weighted Hadamard 1 WC H WH WC 2I WH H WC N N N N/2 2 N N N N 1 1 1 1 1 WC N WC N / 2 I 2 WH NWC H 2 N N N * Moon Ho Lee, “Center Weighted Hadamard Transform” IEEE Trans. on CAS, vol.26, no.9, Sept. 1989 * Moon Ho Lee, and Xiao-Dong Zhang,“Fast Block Center Weighted Hadamard Transform” IEEE Trans. On 8 CAS-I, vol.54, no.12, Dec. 2007. 9 10 11 12 13 14 1 1 1 1 1 m(n ) 2 N 1 C k cos 2 1 1 1 j j 1 X (n) x(k)wnk N m,n m N N [H ]2 [J]4 k0 1 1 1 j j 1 j 2 / N n 0,1...N 1, w e m,n 0,1,..., N 1 1 1 1 1 [H ]n [H ]n / 2 [H ]2 [J]n [J ]n / 2 [H]2 n 4 n1 n1 i j N 3 N 4 k k ik jk (1) k 0 (1) k 0 w(in2 in1 )( jn2 jn1 ) 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 3 5 7 C C C C 1 w w 1 C4 8 8 8 8 1 1 1 1 2 J 4 F 1 w w 2 6 6 2 H 4 3 C8 C8 C8 C8 1 w w 1 1 1 1 1 1 w2 w C 3 C 7 C 1 C 5 8 8 8 8 1 1 1 1 1 1 1 1 i2 / 3 i w e C i cos w1: Hadamard 8 8 w2: Center Weighted Hadamard 1 C 1 C 2 C 3 2 8 8 8 1 1 1 1 1 1 1 1 3 6 7 1 1 1 1 C 8 C 8 C 8 1 1 1 2 1 1 C 2 1 1 1 1 1 1 F3 1 w w 4 1 1 1 1 1/w 1/w 1 2 C 5 C 6 C 1 H4 J 3 2 1 8 8 8 4 1 w w 2 4 1 1 1 1 4 1 1/w 1/w 1 1 C 7 C 2 C 5 8 8 8 1 1 1 1 1 1 1 1 2 i2 /3 j e 6 2 i2 cos cos 1 e 8 8 1 1 1 2 6 cos i2 /3 cos j e 8 8 DFTN DFTN DFT2N DCT N DCT N DCT 2 N H N H N H 2N J N J N J 2 N 2n p 2n 2n ,4n Arbitrary 15 Jacket Definition: element inverse and transpose 1 T Simple Inverse J L and J mn 1/ Lij mn ij mn mn 1 1 1 1 Examples: 1 1 1 1 i i 1 J 4 J 1 2 1 i i 1 1 1 3 J 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 H 4 1 1 1 1 1 1 1 1 1 1 1 2 J 2 J 1 1 1 1 1 1 1 1 1 1 3 2 1 i i 1 1 J 1 4 1 i i 1 where where 1 0, 2 1 1 2 0, 3 1 1 1 1 1 16 17 18 19 20 21 22 23 24 25 Jacket case [ J ]1 [ J ] 2 1 1 1 1 1 1 1 1 4 0 0 0 1 1 1 1 1 2 0 1 1 1 1 1 1 1 1 1 0 4 0 0 1 a b c 1 1 1 2 1 1 0 2 1 1 1 1 4 1 1 1 1 0 0 4 0 1 1 1 1 1 1 1 1 0 0 0 4 ( a) w re a l 2 1 1 1 1 1 1 1 1 4 0 0 0 1 1 1 1 1 2 1 1 2 2 1 1 1 1 1 1 0 6 2 0 1 2 2 1 1 0 3 1 2 2 1 4 1 1 1 1 0 2 6 0 a b c 2 1 1 1 1 1 1 1 1 0 0 0 4 a b 1, c w ( b) w im a g in a ry j 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 j j 1 1 1 1 1 0 (1 j) (1 j) 0 1 1 1 1 1 2 1 j 1 2 2 1 1 1 j j 1 4 1 1 1 1 0 (1 j) (1 j) 0 1 j 2 1 1 0 1 j 2 2 1 1 1 1 1 1 1 1 0 0 0 1 1 2 1 1 3 1 1 2 2 1 1 1 1 1 6 0 0 2 a c b 1 2 1 1 2 1 1 1 1 1 0 6 2 0 3 2 1 2 1 1 1 3 2 1 1 2 4 1 1 1 1 0 2 6 0 a c 1, b 2 1 2 2 1 1 1 1 1 2 0 0 6 1 2 2 1 1 1 1 1 6 0 0 2 a b c 1 2 1 1 1 3 0 2 2 2 2 1 1 1 1 1 0 8 0 0 4 2 2 2 2 4 1 1 1 1 0 0 8 0 a 1,b c 2 2 2 2 1 1 1 4 1 2 2 1 1 1 1 1 2 0 0 6 2 1 1 2 1 1 1 1 6 0 0 2 a b c 2 1 1 1 3 3 1 4 4 1 1 1 1 1 1 0 10 6 0 1 5 1 4 4 1 4 1 1 1 1 0 6 10 0 a 2,b 1,c 4 1 4 2 1 1 1 5 2 1 1 2 1 1 1 1 2 0 0 6 * Moon Ho Lee, “A New Reverse Jacket Transform and Its Fast Algorithm,” IEEE Trans. On Circuit and System 2, vol. 47, no. 1, Jan. 2000. pp. 39-47 26 1 1 J 1 J2 1 1 1 1 1 1 1 1 4 0 0 0 a b 1 1 1 1 2 0 1 1 1 1 1 1 1 1 1 1 0 4 0 0 1 1 1 c 1 2 1 1 2 2 1 1 0 2 4 1 1 1 1 16 1 1 1 10 0 4 0 1 1 1 1 1 1 1 1 0 0 0 4 w real 2 2 2 2 2 1 1 1 1 4 0 0 0 2 1 1 1 3 0 2 1 1 2 1 1 1 10 3 1 0 1 1 1 1 1 2 1 1 2 3 1 11 2 8 2 1 1 2 16 1 1 1 10 1 3 0 2 2 2 2 1 1 1 1 0 0 0 4 a b c a b 1, w imaginary j j j j j 1 1 1 1 4 j 0 0 0 c w j 1 1 j 1 1 1 1 0 2 j 2 2 j 2 0 j 1 1 1 j 1 0 j 1 1 j 1 1 1 1 0 2 j 2 2 j 2 0 1 1 1 1 j 1 2 j j j j 1 1 1 1 0 0 0 4 j j 1 2( j 1) 4 j 16 j 2 1 1 2 1 1 1 1 3 0 0 1 a c b 1 2 2 1 1 1 1 10 3 1 0 1 1 2 1 1 1 1 3 1 1 1 a c 1, 5 2 1 2 5 1 11 3 8 1 2 2 1 16 1 1 1 10 1 3 0 b 2 2 1 1 2 1 1 1 1 1 0 0 3 2 1 1 2 1 1 1 1 3 0 0 1 a b c 1 1 1 1 1 1 1 10 2 0 0 a 1, 1 2 2 1 1 1 1 4 1 1 1 6 2 1 2 6 1 10 3 8 1 1 1 1 16 1 1 1 10 0 2 0 b c 2 2 1 1 2 1 1 1 1 1 0 0 3 2 4 4 2 1 1 1 1 6 0 0 2 a b c 4 1 1 4 1 1 1 1 0 5 3 0 a 2, 1 4 1 1 1 1 1 5 1 1 1 b 1,c 4 9 1 2 2 9 1 13 3 8 4 1 1 4 16 1 1 1 1 0 3 5 0 2 4 4 2 1 1 1 1 2 0 0 6 27 DFT, Fourier,1822 DCT-II, K.R.Rao, 1974 Wavelet, G.Strang,1996 1 N 1 m() n 2 r r nm C kcos2 , m, n 0,1,..., N 1 X( n ) x ( m ) W , 0 n N 1 Nm, n m A NN 2 m0 T r r IIF 0 2 2 2 1,j 1, 2, ..., N 1 T FF Pr 1 1 1/r 1/ r 4 4 4 w here k 1 IIE2 2 0 2 j ,j 0, N * A 2 F 1 j 2 2 1/r 1/ r E 1 1 2 1 j 1 / 2 1 / 2 O C 2 2 2 A Pi A Pj 4 4 4 4 T CC1 3 1 / 2 1 / 2 R 1 1/1 1/j 1 1 4 4 T III 0 *E 2 2 2 2 1 1 2 2 r M 1/1 1/ j j j C * 2 2 IIA2 2 0 2 2 2 IIIINNNN0 FN 0 0 2 2 2 2 2 F ICIIINNNNN/20 /2 0 /2 0 /2 /2 IIIINNNN/20 IN/2 0 /2 0 /2 /2 N C 0 PrNNNN0 FN 0 WII A r N 0KCDII 0 0 N 1 1 2 2 2 2 2 NNNNN/2 /2 /2 /2 /2 0PiNNNN/2 0 AN/2 0 Pj /2 I /2 I /2 1 1 1 1 1 FPF []r [][][][]CPCP A Pi A Pj N N N N rN N c N NNN N IIII0N / 2 0 NNNN/ 2 X( orI N / 2 ) 0 / 2 / 2 / 2 *Jacket Matrices Based on Common Form:X r N 0Pi 0 Pj I I 28 NNNN/ 2 0 X N / 2 / 2 / 2 / 2 Decomposition DCT DFT.