Jacket matrix

In mathematics, a jacket matrix is a square symmetric matrix

of order n if its entries are non-zero and real, complex, or from a finite field, and where In is the identity matrix, and 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: The jacket matrix is a generalization of the Hadamard matrix; it is a diagonal block-wise inverse matrix.

As shown in the table, i.e. in the series, for example with n=2, forward:

That is, there exists an element-wise inverse.

or more general For m x m matrices,

denotes an mn x mn block diagonal Jacket matrix.

Euler's formula: Therefore, Also, Finally,

are pxp Jacket matrix, then

is a block circulant matrix if and only if

, where rt denotes the reciprocal transpose.

{\displaystyle \mathbf {A} _{0}=\left[{\begin{array}{rrrr}-1&1\\1&1\\\end{array}}\right],}

{\displaystyle \mathbf {A} _{1}=\left[{\begin{array}{rrrr}-1&-1\\-1&1\\\end{array}}\right],}

is given by where U, C, A, G denotes the amount of the DNA nucleobases and the matrix

is the block circulant Jacket matrix which leads to the principle of the Antagonism with Nirenberg Genetic Code matrix.

[1] Moon Ho Lee, "The Center Weighted Hadamard Transform", IEEE Transactions on Circuits Syst.

[2] Kathy Horadam, Hadamard Matrices and Their Applications, Princeton University Press, UK, Chapter 4.5.1: The jacket matrix construction, PP.

[3] Moon Ho Lee, Jacket Matrices: Constructions and Its Applications for Fast Cooperative Wireless Signal Processing, LAP LAMBERT Publishing, Germany, Nov. 2012.

[4] Moon Ho Lee, et.

al., "MIMO Communication Method and System using the Block Circulant Jacket Matrix," US patent, no.

[5] 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 17, 2022.

[Available in Online: https://www.intechopen.com/chapters/81329].