In mathematics, a jacket matrix is a square symmetric matrix A = ( a i j ) {\displaystyle A=(a_{ij})} of order n if its entries are non-zero and real, complex, or from a finite field, and
A B = B A = I n {\displaystyle \ AB=BA=I_{n}}where In is the identity matrix, and
B = 1 n ( a i j − 1 ) T . {\displaystyle \ B={1 \over n}(a_{ij}^{-1})^{T}.}where T denotes the transpose of the matrix.
In other words, the inverse of a jacket matrix is determined by its element-wise or block-wise inverse. The definition above may also be expressed as:
∀ u , v ∈ { 1 , 2 , … , n } : a i u , a i v ≠ 0 , ∑ i = 1 n a i u − 1 a i v = { n , u = v 0 , u ≠ v {\displaystyle \forall u,v\in \{1,2,\dots ,n\}:~a_{iu},a_{iv}\neq 0,~~~~\sum _{i=1}^{n}a_{iu}^{-1}\,a_{iv}={\begin{cases}n,&u=v\\0,&u\neq v\end{cases}}}The jacket matrix is a generalization of the Hadamard matrix; it is a diagonal block-wise inverse matrix.
Motivation
| n | .... −2, −1, 0 1, 2,..... | logarithm |
| 2n | .... 1 4 , 1 2 , {\displaystyle \ {1 \over 4},{1 \over 2},} 1, 2, 4, ... | series |
As shown in the table, i.e. in the series, for example with n=2, forward: 2 2 = 4 {\displaystyle 2^{2}=4} , inverse : ( 2 2 ) − 1 = 1 4 {\displaystyle (2^{2})^{-1}={1 \over 4}} , then, 4 ∗ 1 4 = 1 {\displaystyle 4*{1 \over 4}=1} . That is, there exists an element-wise inverse.
Example 1.
A = [ 1 1 1 1 1 − 2 2 − 1 1 2 − 2 − 1 1 − 1 − 1 1 ] , {\displaystyle A=\left[{\begin{array}{rrrr}1&1&1&1\\1&-2&2&-1\\1&2&-2&-1\\1&-1&-1&1\\\end{array}}\right],} : B = 1 4 [ 1 1 1 1 1 − 1 2 1 2 − 1 1 1 2 − 1 2 − 1 1 − 1 − 1 1 ] . {\displaystyle B={1 \over 4}\left[{\begin{array}{rrrr}1&1&1&1\\[6pt]1&-{1 \over 2}&{1 \over 2}&-1\\[6pt]1&{1 \over 2}&-{1 \over 2}&-1\\[6pt]1&-1&-1&1\\[6pt]\end{array}}\right].}or more general
A = [ a b b a b − c c − b b c − c − b a − b − b a ] , {\displaystyle A=\left[{\begin{array}{rrrr}a&b&b&a\\b&-c&c&-b\\b&c&-c&-b\\a&-b&-b&a\end{array}}\right],} : B = 1 4 [ 1 a 1 b 1 b 1 a 1 b − 1 c 1 c − 1 b 1 b 1 c − 1 c − 1 b 1 a − 1 b − 1 b 1 a ] , {\displaystyle B={1 \over 4}\left[{\begin{array}{rrrr}{1 \over a}&{1 \over b}&{1 \over b}&{1 \over a}\\[6pt]{1 \over b}&-{1 \over c}&{1 \over c}&-{1 \over b}\\[6pt]{1 \over b}&{1 \over c}&-{1 \over c}&-{1 \over b}\\[6pt]{1 \over a}&-{1 \over b}&-{1 \over b}&{1 \over a}\end{array}}\right],}Example 2.
For m x m matrices, A j , {\displaystyle \mathbf {A_{j}} ,}
A j = d i a g ( A 1 , A 2 , . . A n ) {\displaystyle \mathbf {A_{j}} =\mathrm {diag} (A_{1},A_{2},..A_{n})} denotes an mn x mn block diagonal Jacket matrix.
J 4 = [ I 2 0 0 0 0 cos θ − sin θ 0 0 sin θ cos θ 0 0 0 0 I 2 ] , {\displaystyle J_{4}=\left[{\begin{array}{rrrr}I_{2}&0&0&0\\0&\cos \theta &-\sin \theta &0\\0&\sin \theta &\cos \theta &0\\0&0&0&I_{2}\end{array}}\right],} J 4 T J 4 = J 4 J 4 T = I 4 . {\displaystyle \ J_{4}^{T}J_{4}=J_{4}J_{4}^{T}=I_{4}.}Example 3.
e i π + 1 = 0 {\displaystyle e^{i\pi }+1=0} , e i π = cos π + i sin π = − 1 {\displaystyle e^{i\pi }=\cos {\pi }+i\sin {\pi }=-1} and e − i π = cos π − i sin π = − 1 {\displaystyle e^{-i\pi }=\cos {\pi }-i\sin {\pi }=-1} .Therefore,
e i π e − i π = ( − 1 ) ( 1 − 1 ) = 1 {\displaystyle e^{i\pi }e^{-i\pi }=(-1)({\frac {1}{-1}})=1} .Also,
y = e x {\displaystyle y=e^{x}} d y d x = e x {\displaystyle {\frac {dy}{dx}}=e^{x}} , d y d x d x d y = e x 1 e x = 1 {\displaystyle {\frac {dy}{dx}}{\frac {dx}{dy}}=e^{x}{\frac {1}{e^{x}}}=1} .Finally,
A·B = B·A = I
Example 4.
Consider [ A ] N {\displaystyle [\mathbf {A} ]_{N}} be 2x2 block matrices of order N = 2 p {\displaystyle N=2p} [ A ] N = [ A 0 A 1 A 1 A 0 ] , {\displaystyle [\mathbf {A} ]_{N}=\left[{\begin{array}{rrrr}\mathbf {A} _{0}&\mathbf {A} _{1}\\\mathbf {A} _{1}&\mathbf {A} _{0}\\\end{array}}\right],} .If [ A 0 ] p {\displaystyle [\mathbf {A} _{0}]_{p}} and [ A 1 ] p {\displaystyle [\mathbf {A} _{1}]_{p}} are pxp Jacket matrix, then [ A ] N {\displaystyle [A]_{N}} is a block circulant matrix if and only if A 0 A 1 r t + A 1 r t A 0 {\displaystyle \mathbf {A} _{0}\mathbf {A} _{1}^{rt}+\mathbf {A} _{1}^{rt}\mathbf {A} _{0}} , where rt denotes the reciprocal transpose.
Example 5.
Let A 0 = [ − 1 1 1 1 ] , {\displaystyle \mathbf {A} _{0}=\left[{\begin{array}{rrrr}-1&1\\1&1\\\end{array}}\right],} and A 1 = [ − 1 − 1 − 1 1 ] , {\displaystyle \mathbf {A} _{1}=\left[{\begin{array}{rrrr}-1&-1\\-1&1\\\end{array}}\right],} , then the matrix [ A ] N {\displaystyle [\mathbf {A} ]_{N}} is given by
[ A ] 4 = [ A 0 A 1 A 0 A 1 ] = [ − 1 1 − 1 − 1 1 1 − 1 1 − 1 1 − 1 − 1 1 1 − 1 1 ] , {\displaystyle [\mathbf {A} ]_{4}=\left[{\begin{array}{rrrr}\mathbf {A} _{0}&\mathbf {A} _{1}\\\mathbf {A} _{0}&\mathbf {A} _{1}\\\end{array}}\right]=\left[{\begin{array}{rrrr}-1&1&-1&-1\\1&1&-1&1\\-1&1&-1&-1\\1&1&-1&1\\\end{array}}\right],} , [ A ] 4 {\displaystyle [\mathbf {A} ]_{4}} ⇒ [ U C A G ] T ⊗ [ U C A G ] ⊗ [ U C A G ] T , {\displaystyle \left[{\begin{array}{rrrr}U&C&A&G\\\end{array}}\right]^{T}\otimes \left[{\begin{array}{rrrr}U&C&A&G\\\end{array}}\right]\otimes \left[{\begin{array}{rrrr}U&C&A&G\\\end{array}}\right]^{T},}where U, C, A, G denotes the amount of the DNA nucleobases and the matrix [ A ] 4 {\displaystyle [\mathbf {A} ]_{4}} is the block circulant Jacket matrix which leads to the principle of the Antagonism with Nirenberg Genetic Code matrix.
[1] Moon Ho Lee, "The Center Weighted Hadamard Transform", IEEE Transactions on Circuits Syst. Vol. 36, No. 9, PP. 1247–1249, Sept. 1989.
[2] Kathy Horadam, Hadamard Matrices and Their Applications, Princeton University Press, UK, Chapter 4.5.1: The jacket matrix construction, PP. 85–91, 2007.
[3] Moon Ho Lee, Jacket Matrices: Constructions and Its Applications for Fast Cooperative Wireless Signal Processing, LAP LAMBERT Publishing, Germany, Nov. 2012.
[4] Moon Ho Lee, et. al., "MIMO Communication Method and System using the Block Circulant Jacket Matrix," US patent, no. US 009356671B1, May, 2016.
[5] S. K. Lee and M. H. Lee, “The COVID-19 DNA-RNA Genetic Code Analysis Using Information Theory of Double Stochastic Matrix,” IntechOpen, Book Chapter, April 17, 2022. [Available in Online: https://www.intechopen.com/chapters/81329].