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 In 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
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^T_4 J_4 =J_4 J^T_4=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.