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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]] | |||
:<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 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]],also it is a [[Diagonal]] block-wise inverse matrix. | |||
== 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.== | |||
:<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> | |||
== References == | |||
* Moon Ho Lee,The Center Weighted Hadamard Transform, ''IEEE Transactions on Circuits'' Syst. Vol. 36, No. 9, PP. 1247–1249, Sept.1989. | |||
* K.J. Horadam, Hadamard Matrices and Their Applications, Princeton University Press, UK, Chapter 4.5.1: The jacket matrix construction, PP. 85–91, 2007. | |||
* Moon Ho Lee, Jacket Matrices: Constructions and Its Applications for Fast Cooperative Wireless Signal Processing,LAP LAMBERT Publishing, Germany,Nov. 2012. | |||
==External links== | |||
* [http://mdmc.chonbuk.ac.kr/english/download/report%201.pdf Technical report: Linear-fractional Function, Elliptic Curves, and Parameterized Jacket Matrices] | |||
* [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.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] | |||
[[Category:Matrices]] |
Revision as of 18:44, 2 March 2013
In mathematics, a jacket matrix is a square matrix of order n if its entries are non-zero and real, complex, or from a finite field, and
where In is the identity matrix, and
where T denotes the transpose of the matrix.
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:
The jacket matrix is a generalization of the Hadamard matrix,also it is a Diagonal block-wise inverse matrix.
Example 1.
or more general
Example 2.
References
- Moon Ho Lee,The Center Weighted Hadamard Transform, IEEE Transactions on Circuits Syst. Vol. 36, No. 9, PP. 1247–1249, Sept.1989.
- K.J. Horadam, Hadamard Matrices and Their Applications, Princeton University Press, UK, Chapter 4.5.1: The jacket matrix construction, PP. 85–91, 2007.
- Moon Ho Lee, Jacket Matrices: Constructions and Its Applications for Fast Cooperative Wireless Signal Processing,LAP LAMBERT Publishing, Germany,Nov. 2012.