# 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 |
| 2$^n$ | ....$\ {1 \over 4},{1 \over 2},$ 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.

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