Jacket matrix

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In mathematics, a jacket matrix is a square matrix of order n if its entries are non-zero and real, complex, or from a finite field, and

Hierarchy of matrix types

where In is the identity matrix, and

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:

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, 2, 4,..... Series

As shown in Table, i.e. in series, n=2 case, Forward: , Inverse  : , then, .

Therefore, exist an element-wise inverse.

Example 1.[edit]

:

or more general

:

Example 2.[edit]

For m x m matrices,

denotes an mn x mn block diagonal Jacket matrix.

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]