Jacket matrix

In mathematics, a jacket matrix is a square symmetric matrix ${\displaystyle A=(a_{ij})}$ of order n if its entries are non-zero and real, complex, or from a finite field, and

Hierarchy of matrix types

${\displaystyle \ AB=BA=I_{n}}$

where I_n is the identity matrix, and

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

where T denotes the transpose of the matrix.

In other words, the inverse of a jacket matrix is determined its element-wise or block-wise inverse. The definition above may also be expressed as:

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

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

Motivation

n	.... −2, −1, 0 1, 2,.....	logarithm
2n	....${\displaystyle \ {1 \over 4},{1 \over 2},}$  1, 2, 4, ...	series

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

Example 1.

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

or more general

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

Example 2.

For m x m matrices, ${\displaystyle \mathbf {A_{j}} ,}$ 

${\displaystyle \mathbf {A_{j}} =\mathrm {diag} (A_{1},A_{2},..A_{n})}$  denotes an mn x mn block diagonal Jacket matrix.

${\displaystyle J_{4}=\left[{\begin{array}{rrrr}I_{2}&0&0&0\\0&\cos \theta &-\sin \theta &0\\0&\sin \theta &\cos \theta &0\\0&0&0&I_{2}\end{array}}\right],}$  ${\displaystyle \ J_{4}^{T}J_{4}=J_{4}J_{4}^{T}=I_{4}.}$ 

Example 3.

Euler's formula:

${\displaystyle e^{i\pi }+1=0}$ , ${\displaystyle e^{i\pi }=\cos {\pi }+i\sin {\pi }=-1}$  and ${\displaystyle e^{-i\pi }=\cos {\pi }-i\sin {\pi }=-1}$ .

Therefore,

${\displaystyle e^{i\pi }e^{-i\pi }=(-1)({\frac {1}{-1}})=1}$ .

Also,

${\displaystyle y=e^{x}}$ 

${\displaystyle {\frac {dy}{dx}}=e^{x}}$ ,${\displaystyle {\frac {dy}{dx}}{\frac {dx}{dy}}=e^{x}{\frac {1}{e^{x}}}=1}$ .

Finally,

A·B = B·A = I

Example 4.

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

${\displaystyle \mathbf {C} _{N}=\left[{\begin{array}{rr}\mathbf {C} _{0}&\mathbf {C} _{1}\\\mathbf {C} _{1}&\mathbf {C} _{0}\\\end{array}}\right]}$ 

be 2x2 block matrix of order N=2p. If ${\displaystyle [\mathbf {C} _{0}]_{p}}$  and ${\displaystyle [\mathbf {C} _{1}]_{p}}$  are pxp Jacket matrices, then ${\displaystyle [\mathbf {C} ]_{N}}$  is the Jacket matrix if and only if

${\displaystyle \ \mathbf {C} _{0}\mathbf {C} _{1}^{RT}+\mathbf {C} _{1}^{RT}\mathbf {C} _{0}=0.}$ 

where RT is reciprocal transpose.

Example 5.

If p=2, a block circulant Jacket matrix (BCJM) ${\displaystyle \mathbf {C} _{N}}$ is given by

${\displaystyle \mathbf {C} _{4}=\left[{\begin{array}{rr}\mathbf {C} _{0}&\mathbf {C} _{1}\\\mathbf {C} _{1}&\mathbf {C} _{0}\\\end{array}}\right]=\left[{\begin{array}{rrrr}1&1&a&-a\\[6pt]1&-1&-1/a&-1/a\\[6pt]a&-a&1&1\\[6pt]-1/a&-1/a&1&-1\\[6pt]\end{array}}\right]_{a=1}=\left[{\begin{array}{rrrr}1&1&0&0\\[6pt]1&-1&0&0\\[6pt]0&0&1&1\\[6pt]0&0&1&-1\\[6pt]\end{array}}\right]+\left[{\begin{array}{rrrr}0&0&1&-1\\[6pt]0&0&-1&-1\\[6pt]1&-1&0&0\\[6pt]-1&-1&0&0\\[6pt]\end{array}}\right]=\left[{\begin{array}{rrrr}1&1&1&-1\\[6pt]1&-1&-1&-1\\[6pt]1&-1&1&1\\[6pt]-1&-1&1&-1\\[6pt]\end{array}}\right].}$ 

where ${\displaystyle \mathbf {C} _{0}}$  and ${\displaystyle \mathbf {C} _{1}}$  are the Hadamard 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, "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] Moon Ho Lee, "MIMO Communication Method and System using the Block Circulant Jacket Matrix]", US patent, US9356671.

External links

• Technical report: Linear-fractional Function, Elliptic Curves, and Parameterized Jacket Matrices
• Jacket Matrix and Its Fast Algorithms for Cooperative Wireless Signal Processing
• Jacket Matrices: Constructions and Its Applications for Fast Cooperative Wireless Signal Processing