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 n .... −2, −1, 0 1, 2,..... logarithm 2n .... 1, 2, 4, ... series

As shown in the table, i.e. in the series, for example with n=2, forward: , inverse : , then, . That is, there exists an element-wise inverse.

Example 1.

or more general :

Example 2. For m x m matrices,

denotes an mn x mn block diagonal Jacket matrix.

Example 3. Euler's formula: , and . Therefore, .

Also,

,.

Finally,

A·B = B·A = I

Example 4.

Consider be 2x2 block matrices of order . If and are pxp Jacket matrix, then is a block circulant matrix if and only if , where rt denotes the reciprocal transpose.

Example 5. Let and , then the matrix 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.

References

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

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

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

Moon Ho Lee, et al., "MIMO Communication Method and System using the Block Circulant Jacket Matrix," US patent, no. US 009356671B1, May, 2016.

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].

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