Biholomorphism

In mathematics, a jacket matrix is a square matrix $A=(a_{ij})$ of order n if its entries are non-zero and real, complex, or from a finite field, and

Hierarchy of matrix types

$\ AB=BA=I_{n}$

where I_n is the identity matrix, and

$\ B={1 \over n}(a_{ij}^{-1})^{T}.$

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:

$\forall u,v\in \{1,2,\dots ,n\}:~a_{iu},a_{iv}\neq 0,~~~~\sum _{i=1}^{n}a_{iu}^{-1}\,a_{iv}={\begin{cases}n,&u=v\\0,&u\neq v\end{cases}}$

The jacket matrix is a generalization of the Hadamard matrix,also it is a Diagonal block-wise inverse matrix.

Example 1.

$A=\left[{\begin{array}{rrrr}1&1&1&1\\1&-2&2&-1\\1&2&-2&-1\\1&-1&-1&1\\\end{array}}\right],$:$B={1 \over 4}\left[{\begin{array}{rrrr}1&1&1&1\\[6pt]1&-{1 \over 2}&{1 \over 2}&-1\\[6pt]1&{1 \over 2}&-{1 \over 2}&-1\\[6pt]1&-1&-1&1\\[6pt]\end{array}}\right].$

or more general

$A=\left[{\begin{array}{rrrr}a&b&b&a\\b&-c&c&-b\\b&c&-c&-b\\a&-b&-b&a\end{array}}\right],$:$B={1 \over 4}\left[{\begin{array}{rrrr}{1 \over a}&{1 \over b}&{1 \over b}&{1 \over a}\\[6pt]{1 \over b}&-{1 \over c}&{1 \over c}&-{1 \over b}\\[6pt]{1 \over b}&{1 \over c}&-{1 \over c}&-{1 \over b}\\[6pt]{1 \over a}&-{1 \over b}&-{1 \over b}&{1 \over a}\end{array}}\right],$

Example 2.

$\mathbf {J} =\left[{\begin{array}{rrrr}I&0&0&0\\0&c&s&0\\0&-s&c&0\\0&0&0&I\\\end{array}}\right],$ :$\mathbf {J} \mathbf {J} ^{\mathrm {T} }=\mathbf {J} ^{\mathrm {T} }\mathbf {J} =\mathbf {I}$

References

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

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