{{Short description|Square matrix that is a generalization of the Hadamard matrix}} In mathematics, a '''jacket matrix''' is a square symmetric matrix <math>A= (a_{ij})</math> of order ''n'' if its entries are non-zero and real, complex, or from a finite field, and thumb|Hierarchy of matrix types

:<math>\ AB=BA=I_n </math>

where ''I''<sub>''n''</sub> is the identity matrix, and :<math>\ B ={1 \over n}(a_{ij}^{-1})^T.</math>

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:

:<math>\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} </math>

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 |-

| 2<sup>''n''</sup> || ....<math>\ {1 \over 4}, {1 \over 2},</math> 1, 2, 4, ...|| series |- |}

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

== Example 1. ==

:<math> A = \left[ \begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 1 & -2 & 2 & -1 \\ 1 & 2 & -2 & -1 \\ 1 & -1 & -1 & 1 \\ \end{array} \right],</math>:<math>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].</math>

or more general :<math> A = \left[ \begin{array}{rrrr} a & b & b & a \\ b & -c & c & -b \\ b & c & -c & -b \\ a & -b & -b & a \end{array} \right], </math>:<math> 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],</math>

== Example 2. == For m x m matrices, <math> \mathbf {A_j},</math> <math>\mathbf {A_j}=\mathrm{diag}(A_1, A_2,.. A_n )</math> denotes an mn x mn block diagonal Jacket matrix. :<math> 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], </math> <math>\ J^T_4 J_4 =J_4 J^T_4=I_4.</math>

== Example 3. == Euler's formula: :<math>e^{i \pi} + 1 = 0</math>, <math>e^{i \pi} =\cos{ \pi} +i\sin{\pi}=-1</math> and <math>e^{-i \pi} =\cos{ \pi} - i\sin{\pi}=-1</math>. Therefore, :<math>e^{i \pi}e^{-i \pi}=(-1)(\frac{1}{-1})=1</math>.

Also, :<math>y=e^{x}</math> :<math>\frac{dy}{dx}=e^{x}</math>,<math>\frac{dy}{dx}\frac{dx}{dy}=e^{x}\frac{1}{e^{x}}=1</math>.

Finally,

'''A'''·'''B''' = '''B'''·'''A''' = '''I'''

== Example 4. ==

Consider <math>[\mathbf {A}]_N</math> be 2x2 block matrices of order <math>N=2p</math> :<math> [\mathbf {A}]_N= \left[ \begin{array}{rrrr} \mathbf {A}_0 & \mathbf {A}_1 \\ \mathbf {A}_1 & \mathbf {A}_0 \\ \end{array} \right],</math>. If <math>[\mathbf {A}_0]_p</math> and <math>[\mathbf {A}_1]_p</math> are pxp Jacket matrix, then <math>[A]_N</math> is a block circulant matrix if and only if <math>\mathbf {A}_0 \mathbf {A}_1^{rt}+\mathbf {A}_1^{rt}\mathbf {A}_0</math>, where rt denotes the reciprocal transpose.

== Example 5. == Let <math>\mathbf {A}_0= \left[ \begin{array}{rrrr} -1 & 1 \\ 1 & 1\\ \end{array} \right],</math> and <math>\mathbf {A}_1= \left[ \begin{array}{rrrr} -1 & -1 \\ -1 & 1\\ \end{array} \right],</math>, then the matrix <math>[\mathbf {A}]_N</math> is given by :<math> [\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],</math>, :<math>[\mathbf {A}]_4 </math>⇒<math> \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, </math> where ''U'', ''C'', ''A'', ''G'' denotes the amount of the DNA nucleobases and the matrix <math>[\mathbf {A}]_4 </math> 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)