Jacket matrix

In mathematics, a jacket matrix is a square symmetric matrix

$A=(a_{ij})$

of order n if its entries are non-zero and real, complex, or from a finite field, and

Hierarchy of matrix types

$AB=BA=I_n$

where I_n is the identity matrix, and

$B=\frac{1}{n}(a_{ij}^{-1}{)}^T.$

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:

$\mathrm{\forall }u,v\in \{1,2,\dots ,n\}:a_{iu},a_{iv}\ne 0,\sum _{i=1}^n{a}_{iu}^{-1}{a}_{iv}=\{\begin{array}{cc}n,& u=v\\ 0,& u\ne v\end{array}$

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	....$\frac{1}{4},\frac{1}{2},$ 1, 2, 4, ...	series

As shown in the table, i.e. in the series, for example with n=2, forward: $2^2=4$, inverse : $(2^2{)}^{-1}=\frac{1}{4}$, then, $4*\frac{1}{4}=1$. That is, there exists an element-wise inverse.

Example 1.

$A=\left[\begin{array}{cccc}1& 1& 1& 1\\ 1& -2& 2& -1\\ 1& 2& -2& -1\\ 1& -1& -1& 1\end{array}\right],$:$B=\frac{1}{4}\left[\begin{array}{cccc}1& 1& 1& 1\\ 1& -\frac{1}{2}& \frac{1}{2}& -1\\ 1& \frac{1}{2}& -\frac{1}{2}& -1\\ 1& -1& -1& 1\end{array}\right].$

or more general

$A=\left[\begin{array}{cccc}a& b& b& a\\ b& -c& c& -b\\ b& c& -c& -b\\ a& -b& -b& a\end{array}\right],$:$B=\frac{1}{4}\left[\begin{array}{cccc}\frac{1}{a}& \frac{1}{b}& \frac{1}{b}& \frac{1}{a}\\ \frac{1}{b}& -\frac{1}{c}& \frac{1}{c}& -\frac{1}{b}\\ \frac{1}{b}& \frac{1}{c}& -\frac{1}{c}& -\frac{1}{b}\\ \frac{1}{a}& -\frac{1}{b}& -\frac{1}{b}& \frac{1}{a}\end{array}\right],$

Example 2.

For m x m matrices, ${𝐀}_{𝐣},$

${𝐀}_{𝐣}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(A_1,A_2,..A_n)$ denotes an mn x mn block diagonal Jacket matrix.

$J_4=\left[\begin{array}{cccc}{I}_2& 0& 0& 0\\ 0& \cos \theta & -\sin \theta & 0\\ 0& \sin \theta & \cos \theta & 0\\ 0& 0& 0& I_2\end{array}\right],$ $J_4^T{J}_4=J_4{J}_4^T=I_4.$

Example 3.

Euler's formula:

$e^{i\pi }+1=0$, $e^{i\pi }=\cos \pi +i\sin \pi =-1$ and $e^{-i\pi }=\cos \pi -i\sin \pi =-1$.

Therefore,

$e^{i\pi }{e}^{-i\pi }=(-1)(\frac{1}{-1})=1$.

Also,

$y=e^x$

$\frac{dy}{dx}=e^x$,$\frac{dy}{dx}\frac{dx}{dy}=e^x\frac{1}{e^x}=1$.

Finally,

A·B = B·A = I

Example 4.

Consider  $[𝐀{]}_N$ be 2x2 block matrices of order $N=2p$ 

$[𝐀{]}_N=\left[\begin{array}{cc}{𝐀}_0& {𝐀}_1\\ {𝐀}_1& {𝐀}_0\end{array}\right],$.

If $[{𝐀}_0{]}_p$ and $[{𝐀}_1{]}_p$ are pxp Jacket matrix, then $[A{]}_N$ is a block circulant matrix if and only if ${𝐀}_0{𝐀}_1^{rt}+{𝐀}_1^{rt}{𝐀}_0$, where rt denotes the reciprocal transpose.

Example 5.

Let ${𝐀}_0=\left[\begin{array}{cc}-1& 1\\ 1& 1\end{array}\right],$ and ${𝐀}_1=\left[\begin{array}{cc}-1& -1\\ -1& 1\end{array}\right],$, then the matrix $[𝐀{]}_N$ is given by

$[𝐀{]}_4=\left[\begin{array}{cc}{𝐀}_0& {𝐀}_1\\ {𝐀}_0& {𝐀}_1\end{array}\right]=\left[\begin{array}{cccc}-1& 1& -1& -1\\ 1& 1& -1& 1\\ -1& 1& -1& -1\\ 1& 1& -1& 1\end{array}\right],$,

$[𝐀{]}_4$⇒${\left[\begin{array}{cccc}U& C& A& G\end{array}\right]}^T\otimes \left[\begin{array}{cccc}U& C& A& G\end{array}\right]\otimes {\left[\begin{array}{cccc}U& C& A& G\end{array}\right]}^T,$

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