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

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

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.

Consider  ${\displaystyle [\mathbf {A} ]_{N}}$ be 2x2 block matrices of order ${\displaystyle N=2p}$ 

${\displaystyle [\mathbf {A} ]_{N}=\left[{\begin{array}{rrrr}\mathbf {A} _{0}&\mathbf {A} _{1}\\\mathbf {A} _{1}&\mathbf {A} _{0}\\\end{array}}\right],}$.

If ${\displaystyle [\mathbf {A} _{0}]_{p}}$ and ${\displaystyle [\mathbf {A} _{1}]_{p}}$ are pxp Jacket matrix, then ${\displaystyle [A]_{N}}$ is a block circulant matrix if and only if ${\displaystyle \mathbf {A} _{0}\mathbf {A} _{1}^{rt}+\mathbf {A} _{1}^{rt}\mathbf {A} _{0}}$, where rt denotes the reciprocal transpose.

Example 5.

Let ${\displaystyle \mathbf {A} _{0}=\left[{\begin{array}{rrrr}-1&1\\1&1\\\end{array}}\right],}$ and ${\displaystyle \mathbf {A} _{1}=\left[{\begin{array}{rrrr}-1&-1\\-1&1\\\end{array}}\right],}$, then the matrix ${\displaystyle [\mathbf {A} ]_{N}}$ is given by

${\displaystyle [\mathbf {A} ]_{4}=\left[{\begin{array}{rrrr}\mathbf {A} _{0}&\mathbf {A} _{1}\\\mathbf {A} _{0}&\mathbf {A} _{1}\\\end{array}}\right]=\left[{\begin{array}{rrrr}-1&1&-1&-1\\1&1&-1&1\\-1&1&-1&-1\\1&1&-1&1\\\end{array}}\right],}$,

${\displaystyle [\mathbf {A} ]_{4}}$⇒${\displaystyle \left[{\begin{array}{rrrr}U&C&A&G\\\end{array}}\right]^{T}\otimes \left[{\begin{array}{rrrr}U&C&A&G\\\end{array}}\right]\otimes \left[{\begin{array}{rrrr}U&C&A&G\\\end{array}}\right]^{T},}$

where U, C, A, G denotes the amount of the DNA nucleobases and the matrix ${\displaystyle [\mathbf {A} ]_{4}}$ 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

• 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