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.

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

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. 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], \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. 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 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. 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.

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

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