Jacket matrix

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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 In 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[edit]

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[disambiguation needed]: 2^2=4 , Inverse  : (2^2)^{-1}={1 \over 4} , then,   4*{1\over 4}=1.

Therefore, exist an element-wise inverse.

Example 1.[edit]


 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.[edit]

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[edit]

  • 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[edit]