Jacket matrix

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.

Motivation

n	.... -2, -1, 0 1, 2,.....	logarithm
2^n	....$\ {1 \over 4},{1 \over 2},$ 1, 2, 4,.....	Series

As shown in Table, i.e. in series, n=2 case, Forward: $2^2=4$, Inverse  : $(2^2)^{-1}={1 \over 4}$, then, $4*{1\over 4}=1$.

Therefore, exist an element-wise inverse.

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.

For m x m matrices, $\mathbf {A_j},$

$\mathbf {A_j}=diag(A_1, A_2,.. A_n )$ denotes an mn x mn block diagonal Jacket matrix.

$J_4 = \left[ \begin{array}{rrrr} I_2 & 0 & 0 & 0 \\ 0 & cos\theta & -sin\theta & 0 \\ 0 & sin\theta & cos\theta & 0 \\ 0 & 0 & 0 & I_2 \end{array} \right],$ $\ J^T_4 J_4 =J_4 J^T_4=I_4.$

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