Jacket matrix: Difference between revisions

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{{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 number|real]], [[complex number|complex]], or from a [[finite field]], and [[File:Had_otr_jac.png|thumb|Hierarchy of matrix types]]

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

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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} =

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</math>

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

==Motivation==

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<math>\mathbf {A_j}=\mathrm{diag}(A_1, A_2,.. A_n )</math>

denotes an mn x mn [[Block matrix|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 \\

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== Example 4. ==

A block circulant Jacket matrix (BCJM) is defined by [3, 5]

Consider <math>[\mathbf {A}]_N</math> be 2x2 block matrices of order <math>N=2p</math>

:<math>

[\mathbf {C}_{NA}]_N= \left[ \begin{array}{rrrrrr} \mathbf {C}_{0A}_0 & \mathbf {CA}_{1} _1 \\ \mathbf {C}_{1A}_1 & \mathbf {C}_{0A}_0 \\ \end{array} \right],</math>.

be 2x2 block matrix of order N=2p. If <math>[\mathbf {C}_{0A}_0]_p</math> and <math>[\mathbf {C}_{1A}_1]_p </math> are pxp Jacket matricesmatrix, then <math>[\mathbf{C}A]_N </math> is thea Jacketblock [[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.

:<math>\ \mathbf{C}_{0}\mathbf{C}_{1}^{RT}+\mathbf{C}_{1}^{RT}\mathbf{C}_{0}=0. </math>

where RT is 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

If p=2, a block circulant Jacket matrix (BCJM) <math>\mathbf{C}_{N} </math>is given by

:<math>

[\mathbf {C}_{4A}]_4= \left[ \begin{array}{rrrrrr} \mathbf {C}_{0A}_0 & \mathbf {CA}_{1} _1 \\ \mathbf {C}_{1A}_0 & \mathbf {C}_{0A}_1 \\ \end{array} \right]=\left[

=\left[ \begin{array}{rrrr} -1 & 1 & a-1 & -a1\\ 1 & 1 & -1 & 1 \\[6pt] -1 & 1 & -1 & -1/a \\ 1 & 1 & -1/a & 1 \\[6pt \end{array} \right],</math>,

:<math>[\mathbf {A}]_4 </math>⇒<math>

a & -a & 1 & 1 \\[6pt] -1/a & -1/a & 1 & -1\\[6pt] \end{array}

\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>

\right]_{a=1}=\left[

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|Genetic Code]] matrix.

\begin{array}{rrrr} 1 & 1 & 0 & 0 \\[6pt] 1 & -1 & 0 & 0 \\[6pt]

0 & 0 & 1 & 1 \\[6pt] 0 & 0 & 1 & -1\\[6pt] \end{array}

\right]+\left[

\begin{array}{rrrr} 0 & 0 & 1 & -1 \\[6pt] 0 & 0 & -1 & -1 \\[6pt]

1 & -1 & 0 & 0 \\[6pt] -1 & -1 & 0 & 0\\[6pt] \end{array}

\right]=\left[

\begin{array}{rrrr} 1 & 1 & 1 & -1 \\[6pt] 1 & -1 & -1 & -1 \\[6pt]

1 & -1 & 1 & 1 \\[6pt] -1 & -1 & 1 & -1\\[6pt] \end{array}

\right].</math>

where <math> \mathbf{C}_{0} </math> and <math> \mathbf{C}_{1} </math> are the Hadamard matrix.

== References ==

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[3] Moon Ho Lee, ''Jacket Matrices: Constructions and Its Applications for Fast Cooperative Wireless Signal Processing'', LAP LAMBERT Publishing, Germany, Nov. 2012.

[54] Moon Ho Lee, et. al., "MIMO Communication Method and System using the Block Circulant Jacket Matrix],", US patent, US9356671no. US 009356671B1, May, 2016.

[4] Moon Ho Lee, "[http://dx.doi.org/10.5772/intechopen.102342 The COVID-19 DNA-RNA Genetic Code Analysis Using Information Theory of Double Stochastic Matrix]", ''Matrix Theory - Classics and Advances'', edited by Mykhaylo I Andriychuk, IntechOpen, June 2022.

[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].

[5] Moon Ho Lee, "MIMO Communication Method and System using the Block Circulant Jacket Matrix]", US patent, US9356671.

==External links==

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* [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)]]