jacket matrix

{{Short description|Square matrix that is a generalization of the Hadamard 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 File:Had_otr_jac.png

: $\ AB=BA=I_n$

where I_n is the identity matrix, and

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

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:

: $\forall u,v \in \{1,2,\dots,n\}:~a_{iu},a_{iv} \neq 0, ~~~~ \sum_{i=1}^n a_{iu}^{-1}\,a_{iv} =
\begin{cases}
n, & u = v\\
0, & u \neq v
\end{cases}$

The jacket matrix is a generalization of the Hadamard matrix; it is a diagonal block-wise inverse matrix.

Motivation

class="wikitable" style="margin:auto;"
n	.... −2, −1, 0 1, 2,.....	logarithm
2ⁿ	.... $\ {1 \over 4}, {1 \over 2},$ 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.

: $A = \left[ \begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 1 & -2 & 2 & -1 \\ 1 & 2 & -2 & -1 \\ 1 & -1 & -1 & 1 \\ \end{array} \right],$ : $B ={1 \over 4} \left[
\begin{array}{rrrr} 1 & 1 & 1 & 1 \\[6pt] 1 & -{1 \over 2} & {1 \over 2} & -1 \\[6pt]
1 & {1 \over 2} & -{1 \over 2} & -1 \\[6pt] 1 & -1 & -1 & 1\\[6pt] \end{array}
\right].$

or more general

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

Example 2.

For m x m matrices, $\mathbf {A_j},$

$\mathbf {A_j}=\mathrm{diag}(A_1, A_2,.. A_n )$

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:

: $e^{i \pi} + 1 = 0$ , $e^{i \pi} =\cos{ \pi} +i\sin{\pi}=-1$ 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 $[\mathbf {A}]_N$ be 2x2 block matrices of order $N=2p$

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

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

Example 5.

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

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

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

where U, C, A, G denotes the amount of the DNA nucleobases and the matrix $[\mathbf {A}]_4$ 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

[https://web.archive.org/web/20110722132439/http://mdmc.chonbuk.ac.kr/english/download/report%201.pdf Technical report: Linear-fractional Function, Elliptic Curves, and Parameterized Jacket Matrices]
[https://web.archive.org/web/20110722132459/http://mdmc.chonbuk.ac.kr/english/images/Jacket%20matrix%20and%20its%20fast%20algorithm%20for%20wireless%20signal%20processing.pdf Jacket Matrix and Its Fast Algorithms for Cooperative Wireless Signal Processing]
[https://www.researchgate.net/publication/342195507_Jacket_Matrices_-Construction_and_Its_Applications_for_Fast_Cooperative_Wireless_Signal_Processing: Jacket Matrices: Constructions and Its Applications for Fast Cooperative Wireless Signal Processing]

Category:Matrices (mathematics)