In mathematics , a jacket matrix is a square symmetric matrix
A
=
(
a
i
j
)
{\displaystyle A=(a_{ij})}
n if its entries are non-zero and real , complex , or from a finite field , and
Hierarchy of matrix types
A
B
=
B
A
=
I
n
{\displaystyle \ AB=BA=I_{n}}
where I n identity matrix , and
B
=
1
n
(
a
i
j
−
1
)
T
.
{\displaystyle \ 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 its element-wise or block-wise inverse. The definition above may also be expressed as:
∀
u
,
v
∈
{
1
,
2
,
…
,
n
}
:
a
i
u
,
a
i
v
≠
0
,
∑
i
=
1
n
a
i
u
−
1
a
i
v
=
{
n
,
u
=
v
0
,
u
≠
v
{\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}}}
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
4
,
1
2
,
{\displaystyle \ {1 \over 4},{1 \over 2},}
series
As shown in the table, i.e. in the series, for example with n =2, forward:
2
2
=
4
{\displaystyle 2^{2}=4}
inverse :
(
2
2
)
−
1
=
1
4
{\displaystyle (2^{2})^{-1}={1 \over 4}}
4
∗
1
4
=
1
{\displaystyle 4*{1 \over 4}=1}
Example 1.
A
=
[
1
1
1
1
1
−
2
2
−
1
1
2
−
2
−
1
1
−
1
−
1
1
]
,
{\displaystyle 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
4
[
1
1
1
1
1
−
1
2
1
2
−
1
1
1
2
−
1
2
−
1
1
−
1
−
1
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].}
or more general
A
=
[
a
b
b
a
b
−
c
c
−
b
b
c
−
c
−
b
a
−
b
−
b
a
]
,
{\displaystyle 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
4
[
1
a
1
b
1
b
1
a
1
b
−
1
c
1
c
−
1
b
1
b
1
c
−
1
c
−
1
b
1
a
−
1
b
−
1
b
1
a
]
,
{\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],}
Example 2.
For m x m matrices,
A
j
,
{\displaystyle \mathbf {A_{j}} ,}
A
j
=
d
i
a
g
(
A
1
,
A
2
,
.
.
A
n
)
{\displaystyle \mathbf {A_{j}} =\mathrm {diag} (A_{1},A_{2},..A_{n})}
J
4
=
[
I
2
0
0
0
0
cos
θ
−
sin
θ
0
0
sin
θ
cos
θ
0
0
0
0
I
2
]
,
{\displaystyle 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
4
T
J
4
=
J
4
J
4
T
=
I
4
.
{\displaystyle \ J_{4}^{T}J_{4}=J_{4}J_{4}^{T}=I_{4}.}
Example 3.
Euler's formula :
e
i
π
+
1
=
0
{\displaystyle e^{i\pi }+1=0}
e
i
π
=
cos
π
+
i
sin
π
=
−
1
{\displaystyle e^{i\pi }=\cos {\pi }+i\sin {\pi }=-1}
e
−
i
π
=
cos
π
−
i
sin
π
=
−
1
{\displaystyle e^{-i\pi }=\cos {\pi }-i\sin {\pi }=-1}
Therefore,
e
i
π
e
−
i
π
=
(
−
1
)
(
1
−
1
)
=
1
{\displaystyle e^{i\pi }e^{-i\pi }=(-1)({\frac {1}{-1}})=1}
Also,
y
=
e
x
{\displaystyle y=e^{x}}
d
y
d
x
=
e
x
{\displaystyle {\frac {dy}{dx}}=e^{x}}
d
y
d
x
d
x
d
y
=
e
x
1
e
x
=
1
{\displaystyle {\frac {dy}{dx}}{\frac {dx}{dy}}=e^{x}{\frac {1}{e^{x}}}=1}
Finally,
A ·B = B ·A = I
Example 4.
A block circulant Jacket matrix (BCJM) is defined by [3, 5]
C
N
=
[
C
0
C
1
C
1
C
0
]
{\displaystyle \mathbf {C} _{N}=\left[{\begin{array}{rr}\mathbf {C} _{0}&\mathbf {C} _{1}\\\mathbf {C} _{1}&\mathbf {C} _{0}\\\end{array}}\right]}
be 2x2 block matrix of order N=2p. If
[
C
0
]
p
{\displaystyle [\mathbf {C} _{0}]_{p}}
[
C
1
]
p
{\displaystyle [\mathbf {C} _{1}]_{p}}
[
C
]
N
{\displaystyle [\mathbf {C} ]_{N}}
C
0
C
1
R
T
+
C
1
R
T
C
0
=
0.
{\displaystyle \ \mathbf {C} _{0}\mathbf {C} _{1}^{RT}+\mathbf {C} _{1}^{RT}\mathbf {C} _{0}=0.}
where RT is reciprocal transpose.
Example 5.
If p=2, a block circulant Jacket matrix (BCJM)
C
N
{\displaystyle \mathbf {C} _{N}}
C
4
=
[
C
0
C
1
C
1
C
0
]
=
[
1
1
a
−
a
1
−
1
−
1
/
a
−
1
/
a
a
−
a
1
1
−
1
/
a
−
1
/
a
1
−
1
]
a
=
1
=
[
1
1
0
0
1
−
1
0
0
0
0
1
1
0
0
1
−
1
]
+
[
0
0
1
−
1
0
0
−
1
−
1
1
−
1
0
0
−
1
−
1
0
0
]
=
[
1
1
1
−
1
1
−
1
−
1
−
1
1
−
1
1
1
−
1
−
1
1
−
1
]
.
{\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].}
where
C
0
{\displaystyle \mathbf {C} _{0}}
C
1
{\displaystyle \mathbf {C} _{1}}
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 | K.J. 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
![{\displaystyle A=\left[{\begin{array}{rrrr}1&1&1&1\\1&-2&2&-1\\1&2&-2&-1\\1&-1&-1&1\\\end{array}}\right],}](/media/api/rest_v1/media/math/render/svg/d0db861b6f4bf217242d96d618704deb3b87ef19)
![{\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].}](/media/api/rest_v1/media/math/render/svg/7a8cc637797577e763b4f3b81e90622673039c40)
![{\displaystyle A=\left[{\begin{array}{rrrr}a&b&b&a\\b&-c&c&-b\\b&c&-c&-b\\a&-b&-b&a\end{array}}\right],}](/media/api/rest_v1/media/math/render/svg/3686796010048b122d0688b008386a6abfd7de65)
![{\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],}](/media/api/rest_v1/media/math/render/svg/6f6ec5287dc89fda4acd0c8f0f37cd1472fd08d1)
![{\displaystyle 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],}](/media/api/rest_v1/media/math/render/svg/f98a1705edcf4b9200d359ce4a252e471501e052)
![{\displaystyle \mathbf {C} _{N}=\left[{\begin{array}{rr}\mathbf {C} _{0}&\mathbf {C} _{1}\\\mathbf {C} _{1}&\mathbf {C} _{0}\\\end{array}}\right]}](/media/api/rest_v1/media/math/render/svg/c8e596abd09b7cc409bcfa4fa6e0da44d6b60fce)
![{\displaystyle [\mathbf {C} _{0}]_{p}}](/media/api/rest_v1/media/math/render/svg/c9dfe374954ba268d76d11d0870cf969dd25cc17)
![{\displaystyle [\mathbf {C} _{1}]_{p}}](/media/api/rest_v1/media/math/render/svg/56e191c572dbfef8eb40c44d7e43d4ffc80331df)
![{\displaystyle [\mathbf {C} ]_{N}}](/media/api/rest_v1/media/math/render/svg/7bb03bf9b468392ece316e8543d6e875059a40c8)
![{\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].}](/media/api/rest_v1/media/math/render/svg/d1f5f9516f69cd7e7568659ae428dc72fce7e89d)
