Leejacket

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

$\ AB=BA=I_{n}$

where I_n is the identity matrix, and

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

where T denotes the transpose of the matrix.

Properties

Set diagram representing the relationship between jacket matrices, Hadamard matrices, and other matrix families

The jacket matrix is a generalization of the Hadamard matrix. The most important property of a jacket matrix is that its inverse may be determined by its element-wise or block-wise inverse.

There are three main classes of matrices :

1.Orthogonal matrices:

$\forall u,v\in \{1,2,\dots ,n\},u\neq v:\sum _{i=1}^{n}{a_{u,i}.a_{v,i}}=0;~~~\sum _{i=1}^{n}a_{u,i}^{2}=n.$

2.Unitary matrices:

$\forall u,v\in \{1,2,\dots ,n\},u\neq v:~\sum _{i=1}^{n}{a_{u,i}{\overline {a_{v,i}}}}=0;~~~\sum _{i=1}^{n}|a_{u,i}|^{2}=n.$

3.Jacket matrices:

$\forall u,v\in \{1,2,\dots ,n\},u\neq v:~\sum _{i=1}^{n}{a_{u,i} \over a_{v,i}}=0;~~~a_{u,i},a_{v,i}\neq 0.$

$\forall u,v\in \{1,2,\dots ,n\},u=v:~\sum _{i=1}^{n}{a_{u,i} \over a_{v,i}}=n.$

Example

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

or more general

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

References

• M.H. Lee, The Center Weighted Hadamard Transform, IEEE Trans.1989 CAS-36, (9), pp.1247-1249.

• M.H. Lee, A New Reverse Jacket Transform and its Fast Algorithm, IEEE Trans. Circuits Syst.-II , vol 47, pp.39-46, 2000.

• M.H. Lee and B.S. Rajan, A Generalized Reverse Jacket Transform, IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 48 no.7 pp 684-691, 2001.

• J. Hou, M.H. Lee and J.Y. Park, New Polynomial Construction of Jacket Transform, IEICE Trans. Fundamentals, vol. E86-A no. 3, pp.652-659, 2003.

• W.P. Ma and M. H. Lee, Fast reverse Jacket Transform Algorithms, Electronics Letter, vol. 39 no. 18 , 2003.

• 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

• Jacket Matrices: Constructions and Its Applications for Fast Cooperative Wireless Signal Processing

Human Nose Morphology and Air Humidity

Human Nose height and width will vary based on air humidity. In Australia Cairns there is a tree called Strangler Fig. A base of the tree is shaped to maximize an absorption of a moisture and resembles a human nose. By comparing morphology of a nose of Asians against Europeans and Americans, one can see that they noticeably differ from each other: Asians’ noses are statistically smaller.

                       	Asian & Korean	     Europe & Americans

Humidity	                 66.8 %         	≈ 30%

Nose (30 years old)	  Height ≈ 2 cm         Height ≈ 2.6 cm

                           Length ≈ 5 cm	  Length ≈ 5.8 cm

One of the consequences of race-dependent nose morphology is a presence of a substantial amount of strong nasal phones in European languages. To compensate for a low-moisture level, Europeans and Americans tend to drink lots of coffee and beer …

References:

1. United Kingdom, Independent News Paper, 27 Aug 2002.

2. Moon Ho Lee, Goal Gate II, Shina Publications, Korea, 15 July 2006.

E-mail: Professor Moon Ho Lee, [email protected]