Jacket matrix

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

A B = B A = I_{n}

where I_n is the identity matrix, and

B = \frac{1}{n} (a_{i j}^{- 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_{i u}, a_{i v} \neq 0, \sum_{i = 1}^{n} a_{i u}^{- 1} a_{i v} = {\begin{matrix} n, & u = v \\ 0, & u \neq v \end{matrix}

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

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

Example 1.

A = [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & - 2 & 2 & - 1 \\ 1 & 2 & - 2 & - 1 \\ 1 & - 1 & - 1 & 1 \end{matrix}],

:

B = \frac{1}{4} [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & - \frac{1}{2} & \frac{1}{2} & - 1 \\ 1 & \frac{1}{2} & - \frac{1}{2} & - 1 \\ 1 & - 1 & - 1 & 1 \end{matrix}] .

or more general

A = [\begin{matrix} a & b & b & a \\ b & - c & c & - b \\ b & c & - c & - b \\ a & - b & - b & a \end{matrix}],

:

B = \frac{1}{4} [\begin{matrix} \frac{1}{a} & \frac{1}{b} & \frac{1}{b} & \frac{1}{a} \\ \frac{1}{b} & - \frac{1}{c} & \frac{1}{c} & - \frac{1}{b} \\ \frac{1}{b} & \frac{1}{c} & - \frac{1}{c} & - \frac{1}{b} \\ \frac{1}{a} & - \frac{1}{b} & - \frac{1}{b} & \frac{1}{a} \end{matrix}],

Example 2.

For m x m matrices, $𝐀_{𝐣},$

$𝐀_{𝐣} = d i a g (A_{1}, A_{2}, .. A_{n})$ denotes an mn x mn block diagonal Jacket matrix.

J_{4} = [\begin{matrix} I_{2} & 0 & 0 & 0 \\ 0 & \cos θ & - \sin θ & 0 \\ 0 & \sin θ & \cos θ & 0 \\ 0 & 0 & 0 & I_{2} \end{matrix}],

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

Example 3.

Euler's formula:

e^{i π} + 1 = 0

,

e^{i π} = \cos π + i \sin π = - 1

and

e^{- i π} = \cos π - i \sin π = - 1

.

Therefore,

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

.

Also,

y = e^{x}

\frac{d y}{d x} = e^{x}

,

\frac{d y}{d x} \frac{d x}{d y} = e^{x} \frac{1}{e^{x}} = 1

.

Finally,

A·B = B·A = I

Example 4.

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

[𝐀]_{N} = [\begin{matrix} 𝐀_{0} & 𝐀_{1} \\ 𝐀_{1} & 𝐀_{0} \end{matrix}],

.

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

Example 5.

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

[𝐀]_{4} = [\begin{matrix} 𝐀_{0} & 𝐀_{1} \\ 𝐀_{0} & 𝐀_{1} \end{matrix}] = [\begin{matrix} - 1 & 1 & - 1 & - 1 \\ 1 & 1 & - 1 & 1 \\ - 1 & 1 & - 1 & - 1 \\ 1 & 1 & - 1 & 1 \end{matrix}],

,

[𝐀]_{4}

⇒

{[\begin{matrix} U & C & A & G \end{matrix}]}^{T} \otimes [\begin{matrix} U & C & A & G \end{matrix}] \otimes {[\begin{matrix} U & C & A & G \end{matrix}]}^{T},

where U, C, A, G denotes the amount of the DNA nucleobases and the matrix $[𝐀]_{4}$ is the block circulant Jacket matrix which leads to the principle of the Antagonism with Nirenberg Genetic Code matrix.