User: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 In is the identity matrix, and
 * $$\ B ={1 \over n}(a_{i,j}^{-1})^T.$$

where T denotes the transpose of the matrix.

Properties


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]. $$

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, moonho@chonbuk.ac.kr