In mathematics, a jacket matrix is a
of order n if its entries are non-zero and real, complex, or from a finite field, and
Hierarchy of matrix types
![{\displaystyle \ AB=BA=I_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6ede91a2385a312e1d12a5e4205a01494859042)
where In is the identity matrix, and
![{\displaystyle \ B={1 \over n}(a_{ij}^{-1})^{T}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66f9689e01b8c6c7a87a2e79d513a73850df208d)
where T denotes the transpose of the matrix.
In other words, the inverse of a jacket matrix is determined its element-wise or block-wise inverse. The definition above may also be expressed as:
![{\displaystyle \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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf1256a06e532ece8f823c650afed3bd0e60a858)
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.[]
:![{\displaystyle 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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a8cc637797577e763b4f3b81e90622673039c40)
or more general
:![{\displaystyle 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],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f6ec5287dc89fda4acd0c8f0f37cd1472fd08d1)
Example 2.[]
For m x m matrices,
denotes an mn x mn block diagonal Jacket matrix.
![{\displaystyle \ J_{4}^{T}J_{4}=J_{4}J_{4}^{T}=I_{4}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a39f689561f13c51769ce1d38652b359fb84764)
Example 3.[]
Euler's formula:
,
and
.
Therefore,
.
Also,
![{\displaystyle y=e^{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e87885ed8d6af200a283fdf841cbea480eb99dd1)
,
.
Finally,
A·B = B·A = I
Example 4.[]
A block circulant Jacket matrix (BCJM) is defined by [3, 5]
![{\displaystyle \mathbf {C} _{N}=\left[{\begin{array}{rr}\mathbf {C} _{0}&\mathbf {C} _{1}\\\mathbf {C} _{1}&\mathbf {C} _{0}\\\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8e596abd09b7cc409bcfa4fa6e0da44d6b60fce)
be 2x2 block matrix of order N=2p. If
and
are pxp Jacket matrices, then
is the Jacket matrix if and only if
![{\displaystyle \ \mathbf {C} _{0}\mathbf {C} _{1}^{RT}+\mathbf {C} _{1}^{RT}\mathbf {C} _{0}=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab3255a47cc63a6f22be4c9ee9607835d063d3cf)
where RT is reciprocal transpose.
Example 5.[]
If p=2, a block circulant Jacket matrix (BCJM)
is given by
![{\displaystyle \mathbf {C} _{4}=\left[{\begin{array}{rr}\mathbf {C} _{0}&\mathbf {C} _{1}\\\mathbf {C} _{1}&\mathbf {C} _{0}\\\end{array}}\right]=\left[{\begin{array}{rrrr}1&1&a&-a\\[6pt]1&-1&-1/a&-1/a\\[6pt]a&-a&1&1\\[6pt]-1/a&-1/a&1&-1\\[6pt]\end{array}}\right]_{a=1}=\left[{\begin{array}{rrrr}1&1&0&0\\[6pt]1&-1&0&0\\[6pt]0&0&1&1\\[6pt]0&0&1&-1\\[6pt]\end{array}}\right]+\left[{\begin{array}{rrrr}0&0&1&-1\\[6pt]0&0&-1&-1\\[6pt]1&-1&0&0\\[6pt]-1&-1&0&0\\[6pt]\end{array}}\right]=\left[{\begin{array}{rrrr}1&1&1&-1\\[6pt]1&-1&-1&-1\\[6pt]1&-1&1&1\\[6pt]-1&-1&1&-1\\[6pt]\end{array}}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1f5f9516f69cd7e7568659ae428dc72fce7e89d)
where
and
are the Hadamard 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, "The COVID-19 DNA-RNA Genetic Code Analysis Using Information Theory of Double Stochastic Matrix", Book Chapter, IntechOpen, June 2022.
[5] Moon Ho Lee, "MIMO Communication Method and System using the Block Circulant Jacket Matrix]", US patent, US9356671. [*https://patentimages.storage.googleapis.com/cb/46/34/4acf23e5a9b6e1/US9356671.pdf]
External links[]