Jacket matrix explained

A = (a _{ij} ) of order n if its entries are non-zero and real, complex, or from a finite field, and AB = BA = I _{n}

where I_n is the identity matrix, and

B = { 1 \over n}(a ^{-1} _{ij} ) ^{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:

\forall u,v \in \{ 1 , 2 , ... ,n\}:~a _{iu} ,a _{iv} ≠ 0 , ~~~~ \sum ^{n} _{i =1} a ^{-1} _{iu} a _{iv} = \begin{cases} n, & u = v\\ 0 , & u ≠ 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
2^n	....{ 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 , inverse : ( 2 ^{2)} ^{-1} = { 1 \over 4 } , then, 4 *{ 1 \over 4 } =1 . That is, there exists an element-wise inverse.

Example 1.

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 \over 4 } \left[ \begin{array}{rrrr} 1 & 1 & 1 & 1 \\[ 6 pt] 1 & - { 1 \over 2 } & { 1 \over 2 } & -1 \\[ 6 pt] 1 & { 1 \over 2 } & - { 1 \over 2 } & -1 \\[ 6 pt] 1 & -1 & -1 & 1 \\[ 6 pt] \end{array} \right].

or more general

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 \over 4 } \left[ \begin{array}{rrrr} { 1 \over a} & { 1 \over b} & { 1 \over b} & { 1 \over a} \\[ 6 pt] { 1 \over b} & - { 1 \over c} & { 1 \over c} & - { 1 \over b} \\[ 6 pt] { 1 \over b} & { 1 \over c} & - { 1 \over c} & - { 1 \over b} \\[ 6 pt] { 1 \over a} & - { 1 \over b} & - { 1 \over b} & { 1 \over a} \end{array} \right],

Example 2.

For m x m matrices,

A _{j} , A _{j} = diag (A _{1 ,} A _{2,..} A _{n} ) denotes an mn x mn block diagonal Jacket matrix.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 ^{T} _{4} J _{4} = J _{4} J ^{T} _{4= I} _{4.}

Example 3.

Euler's formula

e ^{i} + 1 = 0 , e ^{i} = \cos{ \pi} + i\sin{\pi} =-1 and e ^{-i} = \cos{ \pi} - i\sin{\pi} =-1 .Therefore,e ^{i} e ^{-i} = ( -1 )( (1)/(-1) ) =1 .

Also,

y = e ^{x} (dy)/(dx) = e ^{x} ,(dy)/(dx) (dx)/(dy) = e ^{x} (1)/(e ^{x}) =1 .

Finally,

A·B = B·A = I

Example 4.

Consider

[ A ] _{N} be 2x2 block matrices of order N =2 p [ A ] _{N=} \left[ \begin{array}{rrrr} A _{0} & A _{1} \ A _{1} & A _{0} \ \end{array} \right], .If [ A _{0]} _{p} and [ A _{1]} _{p} are pxp Jacket matrix, then [A] _{N} is a block circulant matrix if and only if A _{0} A ^{rt} _{1} + A ^{rt} _{1} A _{0} , where rt denotes the reciprocal transpose.

Example 5.

Let

A _{0=} \left[ \begin{array}{rrrr} -1 & 1 \ 1 & 1 \ \end{array} \right], and A _{1=} \left[ \begin{array}{rrrr} -1 & -1 \ -1 & 1 \ \end{array} \right], , then the matrix [ A ] _{N} is given by[ A ] _{4=} \left[ \begin{array}{rrrr} A _{0} & A _{1} \ A _{0} & 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} ⇒\left[ \begin{array}{rrrr} U & C & A & G\ \end{array} \right] ^{T ⊗ \left[} \begin{array}{rrrr} U & C & A & G\ \end{array} \right] ⊗ \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} is the block circulant Jacket matrix which leads to the principle of the Antagonism with Nirenberg Genetic Code matrix.

References

[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].

External links

• Technical report: Linear-fractional Function, Elliptic Curves, and Parameterized Jacket Matrices
• Jacket Matrix and Its Fast Algorithms for Cooperative Wireless Signal Processing
• Jacket Matrices: Constructions and Its Applications for Fast Cooperative Wireless Signal Processing