The basic concepts and th eo rems of k -Hermitian Circulant s- Hermitian Circulant and s-k Hermitian Circulant matrices are introduced with examples

1. 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
www.ijmsi.org 22 | Page
k– Hermitian Circulant, s-Hermitian Circulant and s – k
Hermitian Circulant Matrices
Dr. N. Elumalai1
, Mr. K. Rajesh kannan2
,
1
Associate professor, 2
Assistant Professor
1
Department of Mathematics, A.V.C. College (Autonomous), Mannampandal, TamilNadu
2
Department 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
Definition: 1For any given z0, z1, z2, z3,…zn-1 Є C
nxn
the Circulant matrix Z =[Zij] is defined by
Z=




















0...321
3...012
2...101
1...210
zznznzn
znzznzn
znzzzn
znzzz

Definition: 2 A matrix Z Є C
nxn
is said to be Hermitian Circulant matrix if Z = Z*
Example: 1
Z =













211
121
112
ii
ii
ii
Z =













211
121
112
ii
ii
ii
Z* =













211
121
112
ii
ii
ii
Result:
(i) All Hermitian is not a Circulant matrix , 





 1
1
i
i
(ii) All Circulant matrix is not a Hermitian matrix, 







11
11
i
i
Definition: 3 A matrix Z Є C
nxn
is said to be k-Hermitian Circulant matrix if Z = K Z* K
Example: 2
Z=













211
121
112
ii
ii
ii
, Z =













211
121
112
ii
ii
ii
, Z*=













211
121
112
ii
ii
ii
, K =










010
100
001

2. k– Hermitian Circulant, s-Hermitian Circulant and s – k Hermitian Circulant Matrices
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Now, KZ*K=










010
100
001













211
121
112
ii
ii
ii










010
100
001
=













211
121
112
ii
ii
ii
= Z
Result: (i) KZ = ZK (ii) ZK = KZ
(i) KZ =










010
100
001













211
121
112
ii
ii
ii
=













ii
ii
ii
121
211
112
ZK =













211
121
112
ii
ii
ii










010
100
001
=













ii
ii
ii
121
211
112
ZK =













ii
ii
ii
121
211
112
KZ = ZK Similarly ZK = KZ
Theorem: 1 Let Z Є C
nxn
is k-Hermitian Circulant matrix if Z = K Z* K
Proof:
KZ*K = KZK where Z* = Z
= ZK K where KZ= ZK
= Z K K
= Z K2
= Z where K2
= I
Theorem: 2 Let Z Є C
nxn
is k-Hermitian Circulant matrix if Z*= K Z K
Proof: K Z K = K Z K where K = K
= K ZK
= K KZ where ZK = 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 1Z K , Z2*= K 2Z K and 1Z = K Z1*K ,
2Z = K Z2* K . To prove Z1 Z2 is k-Hermitian Circulant matrix
We will show that Z1 Z2 = 12 ZZ = K (Z1 Z2)* K
Now K (Z1 Z2)* K = K Z2*Z1* K
= K [ (K 2Z K)( K 1Z K) ] K where Z1*= K 1Z K , Z2*= K 2Z K
= K2
2Z K2
1Z K2
= 2Z 1Z
= 12 ZZ
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 1Z K , Z2*= K 2Z K and 1Z = K Z1*K ,
2Z = K Z2* K. To prove Z1 + Z2 is k-Hermitian Circulant matrix
We will show that ( 21 ZZ  ) = K (Z1 + Z2)* K
Now K (Z1 + Z2)* K = K (Z1*+ Z2*) K
= K Z1*K + K Z2* K
= 1Z + 2Z
= 21 ZZ 
Theorem: 5 Let Z Є C
nxn
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 C
nxn
is k-Hermitian Circulant matrix Z = K Z* K, Z*= K Z K

3. k– Hermitian Circulant, s-Hermitian Circulant and s – k Hermitian Circulant Matrices
www.ijmsi.org 24 | Page
To Prove KZ is k-Hermitian Circulant matrix
We will show that, KZ = ZK = K (KZ)* K
Now K (KZ)* K = K [K( ZK )K] K
= K2
ZK K2
= ZK
Theorem: 5 Let Z Є C
nxn
is k-Hermitian Circulant matrix then ZZT
is also k-Hermitian Circulant matrix
Proof: Let Z Є C
nxn
is k-Hermitian Circulant matrix Z = K Z* K , Z*= K Z K
To Prove ZZT
is k-Hermitian Circulant matrix .We will show that, ZZT
= ZZ
T
= K (ZZT
)* K
Now K (ZZT
)*K = K [ K ( ZZ
T
) K] K
= K2
( ZZ
T
) K2
= ZZ
T
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
(ii) Z1
T
Z2Z1= Z2
T
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 Z K) where Z*= K Z K
= K (K Z K )
= K (K ZK )
= K (KK Z) = K2
(KZ)
= KZ
(ii) Z*K = ZK
Now, Z*K = (K Z K) K where Z*= K Z K
= (K Z K ) K
= (K ZK ) K
= (KK Z) K = K2
(ZK)
= ZK
(iii) (KZ)* = Z*K
Now, (KZ)* = K ( ZK ) K where Z*= K Z K
= K ( Z K ) K
= K ( Z K) K
= (K Z K) K
= Z*K
(iv) (Z*K)* = KZ
Now, (Z*K)* = (ZK)* where (ii)
= K ( KZ ) K
= K ( K Z ) K
= K (K Z K)
= KZ*=KZ

4. k– Hermitian Circulant, s-Hermitian Circulant and s – k Hermitian Circulant Matrices
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Definition: 4 A matrix Z Є Cnxn
is said to be S-Hermitian Circulant matrix if Z = VZ*V where V is Exchange
matrix
Example: 3
VZ*V=










001
010
100













211
121
112
ii
ii
ii










001
010
100
=













211
121
112
ii
ii
ii
= Z
Result: (i) VZ = ZV (ii) ZV = VZ
(i) VZ =










001
010
100













211
121
112
ii
ii
ii
=













ii
ii
ii
112
121
211
(ii) ZV =













211
121
112
ii
ii
ii










001
010
100
=













ii
ii
ii
112
121
211
, ZV =













ii
ii
ii
112
121
211
 VZ = ZV Similarly ZV = VZ
Theorem: 8 Let Z Є C
nxn
is s-Hermitian Circulant matrix if Z = V Z* V
Proof: VZ*V = V Z V where Z* = Z
= ZV K where VZ= ZV
= Z V V
= Z V2
= Z where V2
= I
= Z =Z* where Z* = Z
Theorem:9 Let Z Є C
nxn
is s-Hermitian Circulant matrix if Z*= V Z V
Proof: V Z V = V Z V where V = V
= V ZV
= V VZ where ZV = 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 1Z V , Z2*= V 2Z V and 1Z = V Z1*V ,
2Z = V Z2* V . To prove Z1 Z2 is s-Hermitian Circulant matrix
We will show that Z1 Z2 = 12 ZZ = V (Z1 Z2)* V
Now V (Z1 Z2)* V = V Z2*Z1* V
= V [(V 2Z V) (V 1Z V)] V where Z1*= V 1Z V , Z2*= V 2Z V
= V2
2Z V2
1Z V2
= 1Z 2Z
= 12 ZZ
= Z1 Z2 where 12 ZZ = 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 1Z V, Z2*= V 2Z V and 1Z = V Z1*V ,
2Z = V Z2* V. To prove Z1 + Z2 is s-Hermitian Circulant matrix
We will show that ( 21 ZZ  ) = V (Z1 + Z2)* V
Now V (Z1 + Z2)* V = V (Z1*+ Z2*) V
= V Z1*V + V Z2* V

5. k– Hermitian Circulant, s-Hermitian Circulant and s – k Hermitian Circulant Matrices
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= 1Z + 2Z
= 21 ZZ 
Theorem: 12 Let Z Є C
nxn
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 Z V
To Prove VZ is s-Hermitian Circulant matrix
We will show that, VZ = ZV = V (VZ)* V
Now V (VZ)* V = V [V ( ZV ) V] V
= V2
ZV V2
= ZV = VZ
Theorem: 13 Let Z Є C
nxn
is s-Hermitian Circulant matrix then ZZT
is also s-Hermitian Circulant matrix
Proof: Let Z Є C
nxn
is s-Hermitian Circulant matrix Z = V Z* V , Z*= V Z V
To Prove ZZT
is s-Hermitian Circulant matrix .We will show that, ZZT
= ZZ
T
= V (ZZT
)* V
Now V (ZZT
)*V = V [V ( ZZ
T
) V] V
= V2
( ZZ
T
) V2
= ZZ
T
= Z ZT
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
(ii) Z1
T
Z2Z1 and Z2
T
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 Z V) where Z*= V Z V
= V (V Z V )
= V (V ZV )
= V (VV Z) = V2
(VZ)
= VZ
(ii) Z*V = ZV
Now, Z*V = (V Z V) V where Z*= V Z V
= (V Z V ) V
= (V ZV ) V
= (VV Z) V = V2
(ZV)
= ZV
(iii) (VZ)* = Z*V
Now, (VZ)* = V ( ZV ) V where Z*= V Z V
= V ( Z V ) V
= V ( Z V) V
= (V Z V) V
= Z*V
(iv) (Z*V)* = VZ
Now, (Z*V)* = (ZV)* where (ii)
= V (VZ ) V

6. k– Hermitian Circulant, s-Hermitian Circulant and s – k Hermitian Circulant Matrices
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= V (V Z ) V
= V (V Z 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) Z =KV Z VK (ii) Z =VK Z 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 1Z VK, Z2*=KV 2Z 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 ( 21 ZZ  ) 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 1Z VK, Z2*= KV 2Z 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 ( 21 ZZ ) K
= (Z1Z2)*
Theorem: 17 Let Z Є C
nxn
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 C
nxn
is s-k Hermitian Circulant matrix then ZZT
is also s-k Hermitian Circulant matrix
Proof: Let Z Є C
nxn
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
= K ( ZZ
T
) K
= ( ZZ
T
)*
Theorem: 19 Let Z Є C
nxn
is s-k Hermitian Circulant matrix then the following conditions are equal
(i)Z* =KVZ*VK
(ii) Z* =VKZ*KV
(iii) Z =KV Z VK
(iv) Z =VK Z KV
Proof:
(i) KVZ*VK = K (V Z* V) K
= K Z K
= Z*
(ii) VKZ*KV = V (K Z* K) V
= V Z V
= Z*
(iii) KV Z VK = K [V Z V] K
= K Z*K

7. k– Hermitian Circulant, s-Hermitian Circulant and s – k Hermitian Circulant Matrices
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= Z
(iv) VK Z KV =V (K Z K) V
= V Z* V
= Z
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[4]. Hill, R.D, and Waters, S.R., On k-real and k-hermitian matrices; Lin. Alg. Appl., 169, 17-29 (1992).
[5]. Hazewinkel, Michiel, ed.(2001), ”Symmetric matrix”, Encyclopedia of Mathematics, Springer, ISBN978-1- 55608-010-4
[6]. Krishnamoorthy. S, Vijayakumar. R, Ons-normal matrices, Journal of Analysis and Computation, Vol 15, No 2, (2009).
[7]. Krishnamoorthy. S, Gunasekaran. K, Mohana. N, ”Characterization and theorems on doubly stochastic matrices” Antartica Journal
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