Jacket matrix

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

\ AB=BA=I_{n}

where I_n is the identity matrix, and

\ 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 by its element-wise or block-wise inverse. The definition above may also be expressed as:

\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	.... $\ {1 \over 4},{1 \over 2},$ 1, 2, 4, ...	series

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

Example 1.

A=\left[{\begin{array}{rrrr}1&1&1&1\\1&-2&2&-1\\1&2&-2&-1\\1&-1&-1&1\\\end{array}}\right],

:

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

A=\left[{\begin{array}{rrrr}a&b&b&a\\b&-c&c&-b\\b&c&-c&-b\\a&-b&-b&a\end{array}}\right],

:

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, $\mathbf {A_{j}} ,$

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

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],

\ J_{4}^{T}J_{4}=J_{4}J_{4}^{T}=I_{4}.

Example 3.

Euler's formula:

e^{i\pi }+1=0

,

e^{i\pi }=\cos {\pi }+i\sin {\pi }=-1

and

e^{-i\pi }=\cos {\pi }-i\sin {\pi }=-1

.

Therefore,

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

.

Also,

y=e^{x}

{\frac {dy}{dx}}=e^{x}

,

{\frac {dy}{dx}}{\frac {dx}{dy}}=e^{x}{\frac {1}{e^{x}}}=1

.

Finally,

A·B = B·A = I

Example 4.

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

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

.

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

Example 5.

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

[\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],

,

[\mathbf {A} ]_{4}

⇒

\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 $[\mathbf {A} ]_{4}$ 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