Jacket matrix
In mathematics, a jacket matrix is a square symmetric matrix of order n if its entries are non-zero and real, complex, or from a finite field, and
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
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.
- :
or more general
- :
Example 2.
For m x m matrices,
denotes an mn x mn block diagonal Jacket matrix.
Example 3.
Euler's Formula:
- , and .
Therefore,
- .
Also,
,.
Finally,
A·B=B·A=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
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