Journal of Kerbala University , Vol. 12 No.1 Scientific . 2014

On Near Complex Hadamard Matrices ﺣﻮل اﻟﻤﺼﻔﻮﻓﺎت اﻟﻤﻌﻘﺪة اﻟﻤﻘﺘﺮﺑﺔ ﻣﻦ ﻣﺼﻔﻮﻓﺔ ھﺎدﻣﺮد

Assistant lecturer Rasha Najah Mirzah Department of Mathematics Faculty of Mathematics and Computer Sciences\Universty of Kufa Email: [email protected]

Abstract: In this paper we have introduced a with special condition, and called it "near complex " and we have studied some basic properties of this matrix, also we have found two types of this matrix and conclude some relationships between them.

اﻟﻤﺴﺘﺨﻠﺺ : ﻓﻲ ھﺬا اﻟﺒﺤﺚ ﻗﺪﻣﻨﺎ ﻣﺼﻔﻮﻓﺔ ﺟﺪﯾﺪة ﻣﻊ ﺷﺮوط ﺧﺎﺻﺔ ﺗﺴﻤﻰ " اﻟﻤﺼﻔﻮﻓﺔ اﻟﻤﻌﻘﺪة اﻟﻤﻘﺘﺮﺑﺔ ﻣﻦ ﻣﺼﻔﻮﻓﺔ ھﺎدﻣﺮد " و درﺳﻨﺎ ﺑﻌﺾ ﺑﻌﺾ اﻟﺨﻮاص اﻟﻤﮭﻤﺔ ﻟﮭﺬه اﻟﻤﺼﻔﻮﻓﺔ ﻛﻤﺎ و ﺟﺪﻧﺎ ﻧﻮﻋﯿﻦ ﻣﻨﮭﺎ و اﺳﺘﻨﺘﺠﻨﺎ ﻋﻼﻗﺎت ﻣﮭﻤﺔ ﺑﯿﻨﮭﺎ .

Introduction: A Hadamared matrices seems such simple matrix structures, they are square matrix with entries equal to ±1 whose rows and columns are orthogonal. In other words a Hadamard matrix of order n is a{1,-1}matrix A satisfying: AA = nI where I is the [3] these matrices yet have been actively studied for over 146T years and still have more secrets to be discovered [2]. A Hadamared matrices first introduced by Turyn [6] and further discussed by Seberry Wallis [4],this matrix have very important applications in image processing[5][7] . In 1867 Sylvester[3] proposed a recurrent method for construction of Hadamard matrices of order 2 . In 1893 Hadamard proved this famous determinantal inequality for a positive semi define matrix A,k such that det(A) ≤ h(A) where h(A) is the product of the diagonal entries of A . it follows from this inequality that if A=(a ) is real matrix of order n Which a ≤ 1 then |det A| ≤ n ,This result gives rise to the term n ij "Hadamard matrix". 2 ij A complex Hadamard matrix , h�as �elements 1,-1,i,-i where i= 1 of order 1 or 2 and satisfies CC =2CI , where C denotes to the Hermition conjugates (transpose ,complex conjugates) − of . ∗ ∗ √ 2c

1.Near Complex Hadamard Matrix Definition (1.1) :[3] square matrix of size nxn with entries ±1 is a Hadamard matrix if HH = n. I ; where I is T the identity n × n matrix Hadamard matrices of size n exist only if is two or a multiplen of four.n A Hadamard matrix is said to be skew if H +H = 2I, where denotes the identity matrix. T Definition (1.2): A near complex hadamard matrix is a square complex matrix with entry or – i satisfying : 1. A .A = n.I 2. ItsT rank ; n = 2 ; is any positive integer − n 3. det( ) = n / ;nk = 2 ; is any positive integer n 2 k 192 Journal of Kerbala University , Vol. 12 No.1 Scientific . 2014

Notation(1.3): We will denote to the near complex hadamard matrix by N.C.H. Matrix. Example (1.4): i i i i For instance the matrices A= and B= are two near complex hadamard matrices . i i i i Remark (1.5): − � � � � We can write the N.C.H. Matrix −A of rank 2 as ablock matrix of tensor +(kronker) product of A as follows: k k−1 2 A k −1 A k −1  − A k −1 A k −1  = 2 2 = 2 2 A2k   or A2k   − − − − − A2k 1 A2k 1   A2k 1 A2k 1  Examples (1.6):

i i let A = then i i 1 i i i i 2 � � −A A i i i i 2 1. 4 = AR2 = …1 . …1 = … … ⁞. … … A2 ⋮ A2 ⎡ ⎤ ⎢ i −i ⁞ i −i⎥ � 1 1� ⎢ i i i i ⎥ 2 4/22 2 det ( 2 ) = (4) =16⋮ − ⎢ ⁞ − − ⎥ ⎣ i − i ⁞ i− i ⎦ i i i i i i i i i i i i i i i i ⋮ i i i i ⎡ ⎤ A A i −i i −i ⋮ i −i i −i 3 ⎢ ⎥ 2. AR8 = ( 2 ) = 2 . 2 P = − − ⋮. − − 2 ⋮ 2 ⎢ ⎥ A A ⎢ i −i −i i ⋮ i −i −i i⎥ � ⋯2 ⋯ 2� ⋯i ⋯i ⋯i ⋯i ⋯i ⋯i ⋯i ⋯i 2 2 ⎢ ⎥ ⋮ − ⎢ i i i i ⋮ −i −i −i −i ⎥ ⎢ i −i i −i ⋮ −i i −i i⎥

8/2 4 ⎢ − − ⋮ − − ⎥ 3. det( 8 ) = (8) = 8 = 4096 ⎣ − − ⋮ − − ⎦

Lemma(1.7): If A is near complex hadamard matrix then A is . Proof : its clear from Remark (1.5) that N.C.H matrix is must be symmetric to be N.C.H matrix .

Lemma(1.8): If is near complex hadamard matrix and In is identity matrix of rank n then: 2 1. AP = n I 2. A = ( n) I n 3. A2s − = ( ns ) n Where2s+ 1 s −is any spositive integer Proof: − 1. From definition of near complex hadamard matrix, we get that

AA = n I and since = A P (from lemma 1.7) T T A P = A A − n 2 = A A P = n TI So, A = n I n 2 − − n 193 Journal of Kerbala University , Vol. 12 No.1 Scientific . 2014

2. Its generalized (extension) to the hold (1) in this lemma

A = A P A P 4 = 2n I 2 n.I = ( n) I n n So by using the same way− can 2find− , n − 8 10 A P =( n) P I ,…., A , A , ….. etc.

Thus we 6conclude3 that A = ( n) P I n − 2s s n 3. A P = A P (from A P = (− n) I ) 2s + 1 = ( 2sn) 2s s − n = ( n)s 퐧 − 퐬 퐈 퐀 Lemma(1.9):− 퐀 If A and B are two near complex hadamard matrices ,then : 1. trace (A)= trace (B)=0 2. = where is the conjugate of the matrix A 3. AB= -BA proof퐀� : −퐀 �퐀��� 2k

− + − − 1. trace(A) = ∑ Aij = A2k 1 ( A2k 1 ) = 0 . i= j 4. Then by using remark (1.5) we get that trace (A)= trace (B)=0 2. since A is N.C.H. Matrix and all it's entry are complex number with zero real parts so it's obviously to see that the conjugate of this matrix is equal to the minus of the matrix A , i.e. = 3. form Remark (1.5) we can conclude that there is two equivalent kind of N.C.H. Matrices are : 퐀� −퐀

A k −1 A k −1  − B k −1 B k −1  = 2 2 = 2 2 A2k   and B2k   − − − − − A2k 1 A2k 1   B2k 1 B2k 1 

And by multiplying them we obtained on the following :

0 2A B A.B k k = where 0 is (2 ×2 ) 2A B k−01 k−1 2 2 And � k−1 k−1 � − 2 02 2A B B. A = ( × ) 2A B k0−1 k−1 k k − 2 2 It clear that2 A.B2 = �- B.Ak − 1. k−1 � 2 2 2.The Inverse of Near complex hadamard matrix Theorem(2.1): If A is near complex hadamard matrix and if B = then B is the inverse of A. −1 푘 Proof: 푘 2 2 퐴 Let = −ퟏ 퐤 퐤 ퟐ Then, 퐁 = ퟐ 퐀( ) −ퟏ 퐤( 퐤 ) 퐤 퐀퐁 =퐀ퟐ ퟐ 퐀ퟐ ……. (1) −ퟏ ퟐ 퐤 퐤 ퟐ 퐀ퟐ 194 Journal of Kerbala University , Vol. 12 No.1 Scientific . 2014

And k since P = -2 I ( from lemma(1.8) number (2)) so, by substitute this in equation (1) , we get that

ퟐ퐤 퐀 ퟐ AB = ( ) −ퟏ 퐤 = I 퐤 , where I is the identity matrix ퟐ −ퟐ 퐈 This implies B is the inverse matrix of A ( i.e. =B)

so, −퐤ퟏ 퐀ퟐ = −ퟏ −ퟏ 퐤 퐤 퐤 Examples(2.2): 퐀ퟐ ퟐ 퐀ퟐ 1.If is . . complex matrix of order 2 then:

퐀 퐍 퐂 = 퐇 −ퟏ −ퟏ 퐀 = ퟐ 퐀ퟐ ퟏ ퟐ 퐢 퐢 ퟐ퐢 � � = ퟏ 퐢 ퟏ−퐢 ퟐ퐢 ퟐ퐢 � ퟏ ퟏ � 2.If is . . ퟐ퐢 complex− ퟐ퐢 4×4 matrix then:

퐀 퐍 =퐂 퐇 ( ) −ퟏ −ퟏ ퟐ ퟐ 퐀 ퟐ 퐀ퟐ ퟏ ퟏ ퟏ ퟏ ퟒ퐢 ퟒ퐢 ퟒ퐢 ퟒ퐢 ⎡ ퟏ ퟏ ퟏ ퟏ ⎤ = 퐢 퐢 퐢 퐢 = ퟏ ⎢ퟒ퐢 − ퟒ퐢 ퟒ퐢 − ퟒ퐢⎥ ퟐ 퐢 −퐢 퐢 −퐢 ퟒ퐢 � � ⎢ ퟏ ퟏ ퟏ ퟏ ⎥ 퐢 퐢 −퐢 −퐢 ⎢ퟒ퐢 ퟒ퐢 − ퟒ퐢 − ퟒ퐢⎥ 퐢 −퐢 −퐢 퐢 ⎢ ퟏ ퟏ ퟏ ퟏ ⎥ Theorem(2.3): ⎣ퟒ퐢 − ퟒ퐢 − ퟒ퐢 ퟒ퐢 ⎦ If is matrix ,then is nonsingular if and only if det( ) Proof: see [1] 퐀 퐧퐱퐧 퐀 퐀 ≠ ퟎ Theorem(2.4): [1] If is nonsingular matrix ,then det( ) = ( ) −ퟏ ퟏ Proof: see퐀 [1]퐧퐱퐧 퐀 퐝퐞퐭 퐀 Corollary(2.5): If A is Near complex hadamard matrices then:

-1 det(A ) = ퟐ풌 Proof: 풌 − ퟐ �ퟐ � By definition of Near complex hadamard matrices det(A)= ( ) ퟐ퐤 So, det(A)≠0 ; that is A is non singular (Theorem(2.3)) 퐤 ퟐ Thus, det(A-1) = ( Theorem(2.4)) ퟐ ( ) ퟏ

퐝퐞퐭 퐀

195 Journal of Kerbala University , Vol. 12 No.1 Scientific . 2014

So, det(A-1) = ( ) ퟏ = 퐝퐞퐭 퐀 ( ퟏ) ퟐ퐤 퐤 ퟐ ퟐ = ퟐ풌 Examples(2.6): 풌 − ퟐ 1.the determine of the matrix in example�ퟐ �(2.2) is ,

det( ) = ( ) ퟏ -1 −ퟐ −ퟏ = 2 ퟏ ퟐ ퟐ 퐀 = ퟐ ퟏ 2.the determine ퟐof the matrix o in example (2.2) is ,

( ) P det( ) = −ퟐퟐ − ퟐퟏ ퟐ ퟐ -2 ퟐ = ( ) = 4 = = . 퐀 ퟐ −ퟒ

ퟐ ퟏ ퟏ ퟐ ퟒ ퟒ ퟏퟔ 3.The Relation Between Two N.C.H Matrices Theorem(3.1): If , are two equivalent near complex hadamard matrix then is skew symmetric matrix . Proof:

퐀 퐁 ( ) = P (since , are symmetric) 퐀퐁 퐓 = 퐓 BA퐓 (since AB = BA from lemma 1.9) This implies퐀푩 , (AB퐁) 퐀 = AB 퐀 퐁 So, AB is skew symmetric−T matrix − − Theorem(3.2): If A, B are two equivalent near complex hadamard matrix then, AB = BA . Proof: T T AB = AB ( B is symmetric matrix ) − T = BA = BA (A is symmetric matrix)

− T Theorem(3.3):− If A, B are two equivalent Near complex hadamard matrices then: 1. AB = BA .

2. A�.B =−A.�B Proof : 1. Since − B =�B then , A.B = A(−B) = �− AB = BA = B(−A) ( − A = A then A = −A ) = − BA 2. = = −A − B � � ��� ̅ � 퐴 퐵= AB퐴 퐵 196 Journal of Kerbala University , Vol. 12 No.1 Scientific . 2014

Corollary (3.4): If A,B are two equivalent Near complex hadamard matrices then A&B are skew normal i.e. A(B ) = B(A ) where A & B are the transpose conjugate of the matrices A and B proof:∗ ∗ ∗ ∗ − (B ) = A(BT ) Since B=B T ∗ A(BT ) = A.B Since (AB = BA) = − BA = B(A ) � − � ∗ − 4. The eigen Value of N.C.H Matrix Theorem(4.1) : If A is N.C.H matrix of rank 2 and if λ = ±( 2) i then λ is the eigen value of A. Proof : we will use the mathematical inductionk for the proof k first we must proof that = ±( 2) i is true√ when k=1 second we let = ±( 2 )ki is true whenk k = n then we must proof that λ= ±휆( 2 )ki√ is true when k = n+1 휆 √ √ a a a) first if k=1 then the N.C.H Matrix will be A= a a where 11 12 a , a are equal to i and if a = i a = i conversely � 21 22� if a = i a = i 12 21 11 푡ℎ푒푛 22 − Thus to11 be 22 we must proof that det( ) = 0 a − 푡aℎ푒푛 a )( a ) a a = 0 so a 휆 푒푖푔푒푛a 푣푎푙푢푒 = 0 푡표 then퐴 ( 휆 퐼 − 퐴 11 12 1. if휆 − a = i & a = i 11 22 12 21 � 21 22� 휆 − 휆 − − 푡ℎ푒푛 )( 휆+− ) ( )( ) = 0 + 1 + 1 = 0 = 2 = ± 2 then ( 11 22 so − 2 2 2. by the휆 − same푖 휆 way푖 − if 푖 a 푖 = i & a휆 = i 푡ℎ푒푛 휆 − 푡ℎ푢푠 휆 √ 푖 then ( + )( ) ( )( ) = 0 so 11 22 + 1 + 1 = 0 =− 2 = ± 2

2 휆 푖 휆 − 푖 − 푖 푖2 k b)휆 we assume that 푡 ℎ푒푛= ±휆( 2 )−i is푡ℎ true푢푠 휆 when √k =푖 n finally, we proof that = ±( 2) i is true when k = n+1 as follows 휆 √ k k 휆 √ n c) = ±( 2)P i = 2 i = ± ( 2 ) ( 2) i n+1 = { ± ( 2 )n i}{±( 2) i} {where = -1} 휆 √ �√ � √ √ from (a)&(b) we get that = ±( 2) i is true when k =2 n+1 √ √ 푖 k Examples (4.2): 휆 √ 1. If A is N.C.H matrix of rank 2 then the eigen value be λ = ± 2 i 2. If A is N.C.H matrix of rank 4 then the eigen value be λ = ±2i 3. If A is N.C.H matrix of rank 8 then the eigen value be λ = ±2√ 2 i

√

197 Journal of Kerbala University , Vol. 12 No.1 Scientific . 2014

Theorem(4.3): If A and B are two N.C.H matrices of rank 2 and λ =±(2) i then λ is the eigen value of AB . k k first we must proof that = ±(2) i is true when n=1 0 a k a , a are equal to 2 or 2 a) first if k=1 then the AB will휆 be AB= a 0 where we 12 must proof that 12 21 � 21 � − det( ) = 0 a so a휆 퐼 − 퐴퐵 = 0 then 12 휆 in two cases if a = 2 & a = 2 or if a = 2 & a = 2 � 21 � then +휆4 = 0 = 4 = ±2 12 21 12 21 2 2 − − second휆 we let =푡 ±ℎ푒푛(2 )k휆i is true− when푡ℎ푢푠 휆 k = n 푖 then we must proof that λ= ±(2)ki is true when k = n+1 휆 b) we assume that = ±(2 )ki is true when k = n finally, we proof that = ±(2) i is true when k = n+1 as follows 휆 k k n c) = ±(2)P i = ±(2휆) i = ± ( 2 ) (2) i n = { ±n+ (1 2) i}{±(2) i} {where = -1} from휆 (a)&(b) we get that = ±(2) i is true when k2 = n+1 k 푖 Examples(4.4): 휆 1. If A and B are two N.C.H matrices of rank 2 then the eigen value be λ = ±2i 2. If A and B are two N.C.H matrices of rank 4 then the eigen value be λ = ±4i 3. If A and B are two N.C.H matrices of rank 8 then the eigen value be λ = ±8i

References:

[1] B. Kolman & D. R. Hill; "Elementary Linear algebra with application"; ninth edition,(2008). [2] B. J. Wysocki, T. A. Wysocki, "On some applications of Hadamard matrices" university of wollon gong copyright Springer .Verlag (221-239),(2005) [3] S.S. Again ,"Hadamard matrices and their applications", LNM 1168.springer .(1985). [4] S. wallis, "Hadamard matrices part IV, combinatories, Room squares ,Sum free sets and Hadamard matrices" ,In; Wallis W.D. penfold street A, Seberry Wallis ,J.ceds ,vol 292,springer-verlag,Berlin Heidelberg ,New York (1972). [5] S.M. Balle ,F.G. gustavson M. Joshi, p. palker "A three –dimensional approach to parallel ".September (1995). [6] R.J.Turyn, "complex Hadamard matrices ,combinatorial structures and their applications" pp.435-437.Gordon & Breach, New York ,(1972). [7] X. Zhan ,"open problems in matrix theory", Department of mathematic East China Normal university .Shanghai Zoozul ,China .ICCM.1-4(2004).

198