Jacket Matrix

In mathematics, a jacket matrix is a square symmetric matrix $A= (a_)$ 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
:$\ B =(a_^)^T.$

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:

:$\forall u,v \in \:~a_,a_ \neq 0, ~~~~ \sum_^n a_^\,a_ = \begin n, & u = v\\ 0, & u \neq v \end$

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

Motivation

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

Example 1.

:
A = \left \begin 1 & 1 & 1 & 1 \\ 1 & -2 & 2 & -1 \\ 1 & 2 & -2 & -1 \\ 1 & -1 & -1 & 1 \\ \end \right:B = \left
\begin 1 & 1 & 1 & 1 \\[6pt 1 & - & & -1 \\[6pt">pt.html" ;"title="
\begin 1 & 1 & 1 & 1 \\[6pt">
\begin 1 & 1 & 1 & 1 \\[6pt 1 & - & & -1 \\[6pt 1 & & - & -1 \\[6pt] 1 & -1 & -1 & 1\\[6pt] \end
\right].

or more general
:$A = \left[ \begin a & b & b & a \\ b & -c & c & -b \\ b & c & -c & -b \\ a & -b & -b & a \end \right],$: B = \left \begin & & & \\[6pt & - & & - \\[6pt">pt.html" ;"title=" \begin & & & \\[6pt"> \begin & & & \\[6pt & - & & - \\[6pt & & - & - \\[6pt] & - & - & \end \right],

Example 2.

For m x m matrices, $\mathbf ,$

$\mathbf =\mathrm(A_1, A_2,.. A_n )$
denotes an mn x mn block diagonal Jacket matrix.
:
J_4 = \left \begin I_2 & 0 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta & 0 \\ 0 & \sin\theta & \cos\theta & 0 \\
0 & 0 & 0 & I_2 \end \right $\ J^T_4 J_4 =J_4 J^T_4=I_4.$

Example 3.

Euler's formula:
:$e^ + 1 = 0$, $e^ =\cos +i\sin=-1$ and $e^ =\cos - i\sin=-1$.
Therefore,
:$e^e^=(-1)(\frac)=1$.

Also,
:$y=e^$
:$\frac=e^$,$\frac\frac=e^\frac=1$.

Finally,

A·B = B·A = I

Example 4.

Consider mathbf N be 2x2 block matrices of order $N=2p$
:
mathbf N= \left \begin \mathbf _0 & \mathbf _1 \\ \mathbf _1 & \mathbf _0 \\ \end \right.
If mathbf _0p and mathbf _1p are pxp Jacket matrix, then N is a block circulant matrix if and only if $\mathbf _0 \mathbf _1^+\mathbf _1^\mathbf _0$, where rt denotes the reciprocal transpose.

Example 5.

Let \mathbf _0= \left \begin -1 & 1 \\ 1 & 1\\ \end \right and \mathbf _1= \left \begin -1 & -1 \\ -1 & 1\\ \end \right, then the matrix mathbf N is given by
:
mathbf 4= \left \begin \mathbf _0 & \mathbf _1 \\ \mathbf _0 & \mathbf _1 \\ \end \right=\left \begin -1 & 1 & -1 & -1\\ 1 & 1 & -1 & 1 \\ -1 & 1 & -1 & -1 \\ 1 & 1 & -1 & 1 \\ \end \right,
: mathbf 4 ⇒
\left \begin U & C & A & G\\ \end \rightT\otimes\left \begin U & C & A & G\\ \end \rightotimes\left \begin U & C & A & G\\ \end \rightT,
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

Moon Ho Lee, "The Center Weighted Hadamard Transform", ''IEEE Transactions on Circuits'' Syst. Vol. 36, No. 9, PP. 1247–1249, Sept. 1989.

Kathy 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.

Moon Ho Lee, et. al., "MIMO Communication Method and System using the Block Circulant Jacket Matrix," US patent, no. US 009356671B1, May, 2016.

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 17th, 2022. vailable 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
Matrices