On Direct Product of Hermitian and Symmetric

On Direct Product of Hermitian and Symmetric Matrices With Application Adil Al-Rammahi 

Abstract— In this paper, it is proved that the direct product preserves the property of Hermitian positive definite matrices. A new approach for encoding information was introduced using directness of symmetric matrices. Keywords—Hermiyian Matrices, Encoding Information. I. INTRODUCTION

H

ERMITIAN (and symmetric) matrices occur in various sciences, for instance, in Lyapunov control[13], polarimatic decomposition[14] preconditioners problems[5], coupled—oscillators networks multiple antenna blockcoded[1] spatial time-frequency distribution, electricmagnetic eigenmode[17], robust stability[6], calculations of the frequency band structure of photonic [12] ,an others. For more recent mathematical studies, Cao (2000) study the regular of invertible Hermitian matrix, Castel (2001) study the parallel two-stage methods and Han (1997) study the block structure of Hermitian matrices. In this paper, it is proved that the direct product of two Hermitian positive definite matrices is Hermitian positive definite Symmetric direct product approach was introduced for cipher theory .A new procedure was programmed for reducing L symmetric matrix into direct product of smaller symmetric submatrics.

R2: when all entries of A are real , then it is called symmetric R3: All eigen values of A are real Definition (3,[11] Given a Hermitian matrix A (aij) of dimension m by m, then A is called Hermitian positive definite if it satisfies the following condition (c1): Det(Ak) > 0 for all k rainging from 1 to m where

 a 11  A   a 21 a  k1

Definition (1,[10] A sequare matrix A = (aij) of complex numbers is called Hermitian if A*=At where A* and At denote the conjugate and transpose of A respectively. In other words, A is Hermitian if AH=A where AH denotes the Hermitian conjugate (A*)t Definition (2,[2] If A and B are m by n and m by n and r by s matrices respectively , then an m.r by n.s matrix C=(cij) is called direct product of A and B and denoted by AB such that cij=aij B Remark(1,[18] Let A be Hermitian matrix then: R1: All diagonal entries of H are real Adil L-Rammahi, Kufa University, Faculty of Mathematics and Computer Science, Department of Mathematics, Njaf, IRQ (phone:+964(0)33219195; B.O. 21 Kufa, e-mail: adilm.hasan@ uokufa.edu.iq).

a 22 a k2

 a 1k  (2.1)   a 2k   a kk 

Theorem (1,[18]) Let A be m by m Hermitian matrix, then the following statements are equivalent: T1) CH AC >0 for all row—vector C=(c1,c2,...cm) T2)  ( A ) > 0 for all  where  denotes eigen value of A T3) condition C1 is satisfied Theorem (2,[2]) Given matrices A and B of dimension n by n and m by m respectively , then eigen values of A B are the m.n numbers i(A). j(B) for i = 1,2,.. .,n and j =1,2,...,m. (2.2)

II. BASIC CONCEPTS Following are basic concept which needed in this paper:

a 12

III. NEW MAIN RESULT: This section was concerned to study the direct product of Hermitian matrices as follows: Given a Hermitian matrices A and B of dimension q by q and t by t respectively. Let C A  B ,for determine the diagonal entries Cvv of C with respect to definition (2), then Cvv=aiibkk; i= 1,2,...,q. k=1,2,…,t. v= 1,2,…,q.t (3.1) Since each of A and B is Hermitian, by R1, the diagonal elements of A and B are all real, that is meaning Cvv are all real. For off—diagonal entries: Let Cuv=aijbkL, then C*uv=(aijbkL)* = a*ij b*kL = atij btkL =(aij bkL)t then C*uv = Ctuv

(3.2)

We are in a position to introduce the following proposition:

For decoding the message by taking program IDP ( which implemented in Appendix) obtaining:

Proposition (1) The direct product of two Hermitian matrices is Hermitian Now given A and B be two positive definite Hermitian matrices then by theorem (1): (A) > 0 ,  (B) >0 (3.3) then by theorem(2): (C) = (A).  (B)

(3.4)

Equation (3.4)implies that (C) > 0

(3.5) Then the message word is ( baghda).

Then when Theorem (1) and definition (3) are taken, one can deduced that C is Hermitian positive definite .Therefore one can say that: Proposition (2) The direct product of two Hermitian positive definite matrices is Hermitian positive definite 3.Spmmetric direct product matrices approach for coding systems Matrices can be used to code and decode message. positive integers from 1 through 26 are arbitrarily assignment to the letters of the alphabet, where the assignment is as follows: a

b

c

1

2

3

…

x

y

z

24

25

26

Both the sender and receiver of message have this same table of correspondence between letters and numbers .then C(P) = P M (4.2) where C is a row — cipher text vector, ρ is a row — plaintext vector and M is invertible matrix [ 6,15,3 ]. in this paper ,each set of six (arbitrarily number) letters of message is taken and then distributed into two matrices , where each matrix is symmetric and containing 2 by 2 entries, and each letters is replaced by its corresponding Finally direct product of these matrices is taken. For instance, the message: (baghda) becomes: To send the word (baghda) , we send 16-8-8-4-2-4-1-56-28-7 .The recipient of the message puts it in matrix form:

IV. CONCLUSION It is proved that when each of A and B be Hermitian positive definite matrix , then the direct product of A and B is Hermitian positive definite. In other hand , it is known that the common method of matrix representation in coding theory ,is concerned on multiplying original message vector by invertible choosing matrix. In this paper ,a new approach was introduced to encode plain text message by using original plain text withought invertible matrix . The presented approach is summerized by following steps: first the letters (corresponding numbers) are partitioned into sets. Second each set constructs a symmetric matrix. Third a direct product of two sequence matrices was computed finally the output upper triangular numbers (letters) represent the code words. APPENDIX Let C be n by n real symmetric matrix where n=r.h, then C may be written as a direct product of two submatrices A and B as follows: Procedure “IDP” Step 1) Input integer numbers r and h Step2) Put n=r.h Step3) Input test matrix C Step4) Divide C into r-submatrices say Wij where each submatrix has h by h entries for all i , j = 1 , 2 , 3 , ... , r Slep5) If matrix is symmetric where i