K Hermitian Circulant Matrices

International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759 www.ijmsi.org Volume 4 Issue 10 || December. 2016 || PP-22-28

k– Hermitian Circulant, s-Hermitian Circulant and s – k Hermitian Circulant Matrices

Dr. N. Elumalai1, Mr. K. Rajesh kannan2, 1Associate professor, 2Assistant Professor 1Department of Mathematics, A.V.C. College (Autonomous), Mannampandal, TamilNadu 2Department of Mathematics, Annai College of Engineering and technology, Kovilacheri, TamilNadu.

ABSTRACT: The basic concepts and theorems of k -Hermitian Circulant s- Hermitian Circulant and s-k Hermitian Circulant matrices are introduced with examples. Keywords: k- Hermitian Circulant matrix, s- Hermitian Circulant matrix and s-k- Hermitian Circulant matrix. AMS Classifications: 15B57, 15B05, 47B15

I. INTRODUCTION The concept of s- Hermitian matrices, k- Hermitian matrices and of s-k Hermitian matrices was introduced in [1], [2] and [3] Some properties of Hermitian matrices given in [5],[6] .In this paper, our intention is to define s- Hermitian circulant matrices, k- Hermitian circulant matrices and of s-k Hermitian circulant matrices and prove some results on Hermitian circulant matrices

II. PRELIMINARIES AND NOTATIONS ZT is called Transpose of Z, ZS is called secondary transpose of Z.Let k be a fixed product of disjoint transposition in Sn and ‘K’be the permutation matrix associated with k, V is a permutation matrix with units in the secondary diagonal. Clearly K and V are satisfies the following properties. K2 = I , KT = K,V2 = I , VT = V

III. DEFINITIONS AND THEOREMS nxn Definition: 1For any given z0, z1, z2, z3,…zn-1 Є C the Circulant matrix Z =[Zij] is defined by  z 0 z1 z 2 ... zn  1    zn  1 z 0 z1 ... zn  2 Z=    zn  2 zn  1 z 0 ... zn  3              zn  1 zn  2 zn  3 ... z 0 

Definition: 2 A matrix Z Є Cnxn is said to be Hermitian Circulant matrix if Z = Z* Example: 1  2 1  i 1  i   2 1  i 1  i   2 1  i 1  i        Z = 1  i 2 1  i Z = 1  i 2 1  i Z* = 1  i 2 1  i             1  i 1  i 2  1  i 1  i 2  1  i 1  i 2  Result:  1 i  (i) All Hermitian is not a Circulant matrix ,     i 1  1 1  i  (ii) All Circulant matrix is not a Hermitian matrix,   1  i 1  nxn Definition: 3 A matrix Z Є C is said to be k-Hermitian Circulant matrix if Z = K Z* K Example: 2  2 1  i 1  i   2 1  i 1  i   2 1  i 1  i  1 0 0          Z= 1  i 2 1  i , = 1  i 2 1  i , Z*= 1  i 2 1  i , K = 0 0 1                 1  i 1  i 2  1  i 1  i 2  1  i 1  i 2   0 1 0 

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1 0 0   2 1  i 1  i  1 0 0   2 1  i 1  i          Now, KZ*K= 0 0 1 1  i 2 1  i 0 0 1 = 1  i 2 1  i = Z                  0 1 0  1  i 1  i 2   0 1 0  1  i 1  i 2 

Result: (i) KZ = ZK (ii) ZK = KZ 1 0 0   2 1  i 1  i   2 1  i 1  i        (i) KZ = 0 0 1 1  i 2 1  i = 1  i 1  i 2              0 1 0  1  i 1  i 2  1  i 2 1  i   2 1  i 1  i  1 0 0   2 1  i 1  i   2 1  i 1  i          ZK = 1  i 2 1  i 0 0 1 = 1  i 1  i 2 ZK = 1  i 1  i 2                 1  i 1  i 2   0 1 0  1  i 2 1  i  1  i 2 1  i 

KZ = Similarly ZK = KZ

Theorem: 1 Let Z Є Cnxn is k-Hermitian Circulant matrix if = K Z* K Proof: KZ*K = KZK where Z* = Z

= ZK K where KZ=

= K K = K2 = where K2= I

nxn Theorem: 2 Let Z Є C is k-Hermitian Circulant matrix if Z*= K Z K

Proof: K K = K K where = K = K = K KZ where = KZ = Z =Z* where Z* = Z

Theorem: 3 Let Z1 and Z2 are k-Hermitian Circulant matrices then Z1 Z2 is also k-Hermitian Circulant matrix

Proof: Let Z1 and Z2 are k-Hermitian Circulant matrices then Z1*= K Z 1 K , Z2*= K Z 2 K and Z 1 = K Z1*K ,

Z 2 = K Z2* K . To prove Z1 Z2 is k-Hermitian Circulant matrix

We will show that Z1 Z2 = Z 2 Z 1 = K (Z1 Z2)* K Now K (Z1 Z2)* K = K Z2*Z1* K

= K [ (K Z 2 K)( K K) ] K where Z1*= K K , Z2*= K K = K2 K2 K2 = = Theorem: 4 Let Z1 and Z2 are k-Hermitian Circulant matrices then Z1+ Z2 is also k-Hermitian Circulant matrix

Proof: Let Z1 and Z2 are k-Hermitian Circulant matrices then Z1*= K K , Z2*= K K and = K Z1*K ,

= K Z2* K. To prove Z1 + Z2 is k-Hermitian Circulant matrix

We will show that ( Z 1  Z 2 ) = K (Z1 + Z2)* K Now K (Z1 + Z2)* K = K (Z1*+ Z2*) K = K Z1*K + K Z2* K = + =

Theorem: 5 Let Z Є Cnxn is k-Hermitian Circulant matrix and K is the Permutation matrix , k=(1)(2 3) then KZ is also k-Hermitian Circulant matrix Proof: Let Z Є Cnxn is k-Hermitian Circulant matrix = K Z* K, Z*= K K www.ijmsi.org 23 | Page k– Hermitian Circulant, s-Hermitian Circulant and s – k Hermitian Circulant Matrices

To Prove KZ is k-Hermitian Circulant matrix

We will show that, KZ = ZK = K (KZ)* K Now K (KZ)* K = K [K( )K] K = K2 K2 =

Theorem: 5 Let Z Є Cnxn is k-Hermitian Circulant matrix then ZZT is also k-Hermitian Circulant matrix nxn Proof: Let Z Є C is k-Hermitian Circulant matrix Z = K Z* K , Z*= K K T T T T To Prove ZZ is k-Hermitian Circulant matrix .We will show that, ZZ = Z Z = K (ZZ )* K T T Now K (ZZ )*K = K [ K ( Z Z ) K] K 2 T 2 = K ( Z Z ) K

T = Z Z Result: Let Z1 and Z2 are k-Hermitian Circulant matrices for the following conditions are holds (i) Z1 Z2 = Z2 Z1 and also k-Hermitian Circulant matrix T T (ii) Z1 Z2Z1= Z2 Z1Z2 are also k-Hermitian Circulant matrices

Theorem: 7 Let Z is k-Hermitian Circulant matrix then the following conditions are equal 1) KZ* = KZ 2) Z*K = ZK 3) (KZ)*= Z*K 4) (Z*K) = KZ Proof: (i) KZ* = KZ Now, KZ* = K (K K) where Z*= K K

= K (K K ) = K (K ) = K (KK Z) = K2 (KZ) = KZ (ii) Z*K = ZK Now, Z*K = (K K) K where Z*= K K = (K ) K = (K ) K = (KK Z) K = K2 (ZK) = ZK (iii) (KZ)* = Z*K Now, (KZ)* = K ( ) K where Z*= K K = K ( ) K = K ( K) K = (K K) K = Z*K (iv) (Z*K)* = KZ Now, (Z*K)* = (ZK)* where (ii)

= K ( KZ ) K = K ( ) K = K (K K) = KZ*=KZ

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nxn Definition: 4 A matrix Z Є C is said to be S-Hermitian Circulant matrix if Z = VZ*V where V is Exchange matrix Example: 3  0 0 1   2 1  i 1  i   0 0 1   2 1  i 1  i          VZ*V= 0 1 0 1  i 2 1  i 0 1 0 = 1  i 2 1  i =                 1 0 0  1  i 1  i 2  1 0 0  1  i 1  i 2 

Result: (i) VZ = ZV (ii) ZV = VZ  0 0 1   2 1  i 1  i  1  i 1  i 2        (i) VZ = 0 1 0 1  i 2 1  i = 1  i 2 1  i             1 0 0  1  i 1  i 2   2 1  i 1  i 

 2 1  i 1  i   0 0 1  1  i 1  i 2  1  i 1  i 2          (ii) ZV = 1  i 2 1  i 0 1 0 = 1  i 2 1  i , ZV = 1  i 2 1  i                 1  i 1  i 2  1 0 0   2 1  i 1  i   2 1  i 1  i 

 VZ = ZV Similarly ZV = VZ

Theorem: 8 Let Z Є Cnxn is s-Hermitian Circulant matrix if = V Z* V Proof: VZ*V = V Z V where Z* = Z = K where VZ=

= V V = V2 = where V2= I = Z =Z* where Z* = Z

Theorem:9 Let Z Є Cnxn is s-Hermitian Circulant matrix if Z*= V V

Proof: V V = V V where V = V = V = V VZ where = VZ = V2 Z where V2=I = Z =Z* where Z* = Z

Theorem:10 Let Z1 and Z2 are s-Hermitian Circulant matrices then Z1 Z2 is also s-Hermitian Circulant matrix

Proof: Let Z1 and Z2 are s-Hermitian Circulant matrices then Z1*= V Z 1 V , Z2*= V Z 2 V and = V Z1*V ,

= V Z2* V . To prove Z1 Z2 is s-Hermitian Circulant matrix

We will show that Z1 Z2 = Z 2 Z 1 = V (Z1 Z2)* V Now V (Z1 Z2)* V = V Z2*Z1* V

= V [(V V) (V V)] V where Z1*= V V , Z2*= V V = V2 V2 V2 = =

= Z1 Z2 where = Z1 Z2

Theorem:11 Let Z1 and Z2 are s-Hermitian Circulant matrices then Z1+ Z2 is also s-Hermitian Circulant matrix

Proof: Let Z1 and Z2 are s-Hermitian Circulant matrices then Z1*= V V, Z2*= V V and = V Z1*V ,

= V Z2* V. To prove Z1 + Z2 is s-Hermitian Circulant matrix

We will show that ( Z 1  Z 2 ) = V (Z1 + Z2)* V Now V (Z1 + Z2)* V = V (Z1*+ Z2*) V = V Z1*V + V Z2* V www.ijmsi.org 25 | Page k– Hermitian Circulant, s-Hermitian Circulant and s – k Hermitian Circulant Matrices

= Z 1 + Z 2

= Z 1  Z 2

Theorem: 12 Let Z Є Cnxn is s-Hermitian Circulant matrix and V is the Exchange matrix then VZ is also s- Hermitian Circulant matrix

Proof: Let Z is s-Hermitian Circulant matrix then Z = V Z* V, Z*= V V To Prove VZ is s-Hermitian Circulant matrix

We will show that, VZ = ZV = V (VZ)* V Now V (VZ)* V = V [V ( ) V] V = V2 V2 = = VZ

Theorem: 13 Let Z Є Cnxn is s-Hermitian Circulant matrix then ZZT is also s-Hermitian Circulant matrix Proof: Let Z Є Cnxn is s-Hermitian Circulant matrix = V Z* V , Z*= V V T T T T To Prove ZZ is s-Hermitian Circulant matrix .We will show that, ZZ = Z Z = V (ZZ )* V T T Now V (ZZ )*V = V [V ( Z Z ) V] V 2 T 2 = V ( Z Z ) V T T = Z Z = Z Z Result: Let Z1 and Z2 are s-Hermitian Circulant matrices for the following conditions are holds (i) Z1 Z2 = Z2 Z1 and also s-Hermitian Circulant matrix T T (ii) Z1 Z2Z1 and Z2 Z1Z2 are also s-Hermitian Circulant matrices

Theorem: 14 Let Z is s-Hermitian Circulant matrix then the following conditions are equal 1) VZ* = VZ 2) Z*V = ZV 3) (VZ)*= Z*V 4) (Z*V) = VZ Proof: (i) VZ* = VZ Now, VZ* = V (V V) where Z*= V V

= V (V V ) = V (V ) = V (VV Z) = V2 (VZ) = VZ (ii) Z*V = ZV Now, Z*V = (V V) V where Z*= V V = (V ) V = (V ) V = (VV Z) V = V2 (ZV) = ZV (iii) (VZ)* = Z*V Now, (VZ)* = V ( ) V where Z*= V V = V ( ) V = V ( V) V = (V V) V = Z*V (iv) (Z*V)* = VZ Now, (Z*V)* = (ZV)* where (ii)

= V ( VZ ) V www.ijmsi.org 26 | Page k– Hermitian Circulant, s-Hermitian Circulant and s – k Hermitian Circulant Matrices

= V ( V Z ) V = V (V V) = VZ*=VZ

Definition: 5 A matrix Z Є Cnxn is said to be s-k Hermitian Circulant matrix if (i)Z =KVZ*VK (ii) Z =VKZ*KV (iii) =KV VK (ii) =VK KV where V is Exchange matrix and K is Permutation matrix K = (1)(2 3)

Theorem: 15Let Z1 and Z2 are s-k Hermitian Circulant matrices then Z1+ Z2 is also s-k Hermitian Circulant matrix

Proof: Let Z1 and Z2 are s-k Hermitian Circulant matrices then Z1*= KV Z 1 VK, Z2*=KV Z 2 VK To prove Z1 + Z2 is s-k Hermitian Circulant matrix We will show that (Z1+Z2)* = KV (Z1 + Z2)* VK Now KV (Z1 + Z2)* VK = K [V (Z1*+ Z2*) V] K

= K ( Z 1  Z 2 ) K = (Z1*+ Z2*) Theorem: 16 Let Z1 and Z2 are s-k Hermitian Circulant matrices then Z1 Z2 is also s-k Hermitian Circulant matrix

Proof: Let Z1 and Z2 are s-k Hermitian Circulant matrices then Z1*= KV VK, Z2*= KV VK To prove Z1 Z2 is s-k Hermitian Circulant matrix We will show that (Z1 Z2)* = KV (Z1 Z2)* VK Now KV (Z1 Z2)* VK = K [VZ2*Z1*V] K

= K ( Z 1 Z 2 ) K = (Z1Z2)*

Theorem: 17 Let Z Є Cnxn is s-k Hermitian Circulant matrix and V is the Exchange matrix and K is Permutation matrix K= (1)(2 3) then VZ is also s-k Hermitian Circulant matrix Proof: Let Z is s-k Hermitian Circulant matrix then Z*= KVZ*VK To Prove VZ is s-k Hermitian Circulant matrix We will show that, (VZ)* =KV (VZ)* VK Now KV(VZ)*VK= K [V (VZ)*V] K = K ZV K = (VZ)* Theorem: 18 Let Z Є Cnxn is s-k Hermitian Circulant matrix then ZZT is also s-k Hermitian Circulant matrix Proof: Let Z Є Cnxn is s-k Hermitian Circulant matrix Z*= KVZ*VK To Prove ZZT is s-k Hermitian Circulant matrix .We will show that, (ZZT)* = KV (ZZT)* VK Now KV (ZZT)*VK = K [V (ZZT)* V] K

T = K ( Z Z ) K

T = ( Z Z )* Theorem: 19 Let Z Є Cnxn is s-k Hermitian Circulant matrix then the following conditions are equal (i)Z* =KVZ*VK (ii) Z* =VKZ*KV (iii) =KV VK (iv) =VK KV Proof: (i) KVZ*VK = K (V Z* V) K = K K = Z* (ii) VKZ*KV = V (K Z* K) V = V V = Z* (iii) KV VK = K [V V] K = K Z*K

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= Z

(iv) VK Z KV =V (K K) V = V Z* V = Z

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