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| In [[mathematics]], a '''jacket matrix''' is a [[square 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]]
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| :<math>\ AB=BA=I_n </math>
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| where ''I''<sub>''n''</sub> is the [[identity matrix]], and
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| :<math>\ B ={1 \over n}(a_{ij}^{-1})^T.</math>
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| where ''T'' denotes the [[transpose]] of the matrix.
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| In other words, the inverse of a jacket matrix is determined its element-wise or block-wise inverse. The definition above may also be expressed as:
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| :<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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| \begin{cases}
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| n, & u = v\\
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| 0, & u \neq v
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|
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| \end{cases}
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| </math>
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| The jacket matrix is a generalization of the [[Hadamard matrix]],also it is a [[Diagonal]] block-wise inverse matrix.
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| == Example 1. ==
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| :<math>
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| 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[
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| \begin{array}{rrrr} 1 & 1 & 1 & 1 \\[6pt] 1 & -{1 \over 2} & {1 \over 2} & -1 \\[6pt]
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| 1 & {1 \over 2} & -{1 \over 2} & -1 \\[6pt] 1 & -1 & -1 & 1\\[6pt] \end{array}
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| \right].</math>
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| or more general
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| :<math>
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| A = \left[ \begin{array}{rrrr} a & b & b & a \\ b & -c & c & -b \\ b & c & -c & -b \\
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| 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>
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| == Example 2.==
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| :<math> \mathbf{J}= \left[ \begin{array}{rrrr} I & 0 & 0 & 0 \\ 0 & c & s & 0 \\ 0 & -s & c & 0 \\ 0 & 0 & 0 & I \\ \end{array} \right],</math> :<math> \mathbf{J}\mathbf{J}^{\mathrm{T}} = \mathbf{J}^{\mathrm{T}}\mathbf{J} =\mathbf{I}</math> | |
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| == References ==
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| * Moon Ho Lee,The Center Weighted Hadamard Transform, ''IEEE Transactions on Circuits'' Syst. Vol. 36, No. 9, PP. 1247–1249, Sept.1989.
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| * K.J. Horadam, Hadamard Matrices and Their Applications, Princeton University Press, UK, Chapter 4.5.1: The jacket matrix construction, PP. 85–91, 2007.
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| * Moon Ho Lee, Jacket Matrices: Constructions and Its Applications for Fast Cooperative Wireless Signal Processing,LAP LAMBERT Publishing, Germany,Nov. 2012.
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| ==External links==
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| * [http://mdmc.chonbuk.ac.kr/english/download/report%201.pdf Technical report: Linear-fractional Function, Elliptic Curves, and Parameterized Jacket Matrices]
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| * [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]
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| * [https://www.morebooks.de/store/gb/book/jacket-matrices/isbn/978-3-659-29145-6: Jacket Matrices: Constructions and Its Applications for Fast Cooperative Wireless Signal Processing]
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| [[Category:Matrices]]
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Greetings! I am Myrtle Shroyer. North Dakota is exactly where me and my husband live. Hiring is her day job now and she will not change it whenever soon. He is really fond of doing ceramics but he is having difficulties to discover time for it.
my web page: http://www.gaysphere.net