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Wikipedia / 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.

Motivation

As shown in the table, i.e. in the series, for example with n, inverse. That is, there exists an element-wise inverse.

Example 1.

or more general

Example 2.

denotes an mn x mn block diagonal Jacket matrix.

Example 3.

Euler's formula:

  1. .

Therefore,

  1. .

Also,

  1. .

Finally,

A·B = B·A = I

Example 4.

[\mathbf {A}]_NN=2p

  1. .

is a block circulant matrix, where rt denotes the reciprocal transpose.

Example 5.

is given by

  1. ,

where U, C, A, G is the block circulant Jacket matrix which leads to the principle of the Antagonism with Nirenberg Genetic Code matrix.

References

[1] Moon Ho Lee, "The Center Weighted Hadamard Transform", IEEE Transactions on Circuits Syst. Vol. 36, No. 9, PP. 1247–1249, Sept. 1989.

[2] Kathy Horadam, Hadamard Matrices and Their Applications, Princeton University Press, UK, Chapter 4.5.1: The jacket matrix construction, PP. 85–91, 2007.

[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. US 009356671B1, May, 2016.

[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].