Symmetric Doubly Stochastic Matrices

International Journal of Engineering Science Invention ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726 www.ijesi.org Volume 3 Issue 8 ǁ August 2014 ǁ PP.11-16

k - Symmetric Doubly Stochastic, s - Symmetric Doubly Stochastic and s – k - Symmetric Doubly Stochastic Matrices

1,Dr. K. Gunasekaran , 2,Mrs. N. Mohana, Associate professor1 Assistant Professor2 1Department of Mathematics, Government Arts College (Autonomous), Kumbakonam, TamilNadu. 2 Department of Mathematics, A.V.C. College (Autonomous), Mannampandal, TamilNadu.

ABSTRACT: The basic concepts and theorems of k-symmetric doubly stochastic s-symmetric doubly stochastic and s-k-symmetric doubly stochastic matrices are introduced with examples.

KEY WORDS: k-symmetric doubly stochastic matrix, s-symmetric doubly stochastic matrix and s-k-symmetric doubly stochastic matrix.

AMS CLASSIFICATIONS: 15A51, 15B99

I. INTRODUCTION We have already seen the concept of symmetric doubly stochastic matrices. In this paper the symmetric doubly stochastic matrix is developed in real matrices. Recently Hill and Waters[2] have developed a theory of k-real matrices as a generalization of s-real matrices. Ann Lee[1] has initiated the study of secondary symmetric matrices, that is matrices whose entries are symmetric about the secondary diagonal. Ann Lee[1] has shown that the matrix A, the usual transpose AT and secondary transpose AS are related as AS = VATV and AT = VASV where V is a permutation matrix with units in the secondary diagonal.

II. PRELIMINARIES AND NOTATIONS AT - Transpose of A Let k be a fixed product of disjoint transpositions in Sn and „K‟ be the permutation matrix associated with k. Clearly K satisfies the following properties. K 2 = I, KT = K.

III. DEFINITIONS AND THEOREMS DEFINITION: 1 A matrix A  Rn x n is said to be symmetric doubly stochastic matrix if A = AT and = 1, j = 1, 2, ………n

and = 1, i = 1, 2, ………n and all a ij ≥ 0. If A is doubly stochastic and also symmetric then it is called a symmetric doubly stochastic matrix.

DEFINITION: 2 A matrix A  Rn x n is said to be k-symmetric doubly stochastic matrix if A = K AT K

THEOREM: 1 Let A  Rn x n is k-symmetric doubly stochastic matrix then A = K AT K. Proof: K AT K = KAK where AT = A = AKK where AK = KA = AK2 = A where K2 = I THEOREM: 2 Let AT  Rn x n is k-symmetric doubly stochastic matrix then AT = KAK. Proof: K A K = KATK where A = AT = ATKK where K AT = AT K = AT K2 = AT where K2 = I www.ijesi.org 11 | Page k - Symmetric Doubly Stochastic, s – Symmetric…

THEOREM: 3 Let A, B  Rn x n is k-symmetric doubly stochastic matrix then (A + B) is k-symmetric doubly stochastic matrix. Proof: Let A and B are k-symmetric doubly stochastic matrix if A = K AT K and B = KBTK. To prove (A + B) is k-symmetric doubly stochastic matrix we will show that

(A + B) = K (A + B)T K

Now K (A + B)T K = K (AT + B T) K = K (AT + B T) K = (KAT + KB T) K = (KAT K+ KB T K)

= (A + B) where K AT K = A and KBTK = B THEOREM: 4 Any k-symmetric doubly stochastic matrix can be represent as sum of k-symmetric doubly stochastic matrix and skew k-symmetric doubly stochastic matrix. Proof: To prove that (A + KATK) and (A - KATK) are k-symmetric doubly stochastic matrices the we will show that (A + KATK) = K (A + KATK)T K and (A - KATK) = K (A - KATK)T K.

K (A + KATK)T K = (A + KATK) using theorem 3 and K (A - KATK)T K = (A - KATK).

Then (A + KATK) + (A - KATK) = 2A/2 = A. Hence the theorem is proved.

THEOREM: 5 If A and B are k-symmetric doubly stochastic matrices then AB is also k-symmetric doubly stochastic matrix. Proof: Let A and B are k-symmetric doubly stochastic matrix if A = K AT K and B = KBTK. Since AT and BT are also k-symmetric doubly stochastic matrices then AT = KAK and BT = KBK. To prove A B is k-symmetric doubly stochastic matrix we will show that AB= K (A B)T K Now K (A B)T K= KBTATK = K(KBK)(KAK)K where AT = KAK and BT = KBK. = K2B K2A K2 = BA where K2 = I = AB where BA = AB

THEOREM: 6 If A and B are k-symmetric doubly stochastic matrices and K is the permutation matrix, k = {(1), (2 3)} then KA is also k-symmetric doubly stochastic matrix. Proof: Let A and B are k-symmetric doubly stochastic matrix if A = K AT K and B = KBTK. Since AT and BT are also k-symmetric doubly stochastic matrices then AT = KAK and BT = KBK. To prove K B is K-symmetric doubly stochastic matrix we will show that KA= K (KA)T K Now K (KA)T K= K(ATKT )K = KATKTK = KAT where KT K = I . = KA where KAT = KA

RESULT: For A Rn x n is symmetric doubly stochastic matrices for the following are holds. [1] A = K AT K [2] KA is symmetric doubly stochastic matrix. [3] AK is symmetric doubly stochastic matrix.

Example:

A = AT = and k = (1) (2 3) =

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(i) K AT K = = A

Similarly KAK = AT

(ii) KA = = = (KA)T

 KA is symmetric doubly stochastic matrix. Similarly KAT is also symmetric doubly stochastic matrix.

(iii) AK = = = (AK)T

 AK is symmetric doubly stochastic matrix. Similarly AT K is also symmetric doubly stochastic matrix.

DEFINITION: 3 A matrix A  Rn x n is said to be s-symmetric doubly stochastic matrix if AS = V AT V where V is a permutation matrix with units in the secondary diagonal.

THEOREM: 7 Let A  Rn x n is s-symmetric doubly stochastic matrix then AS = V AT V. Proof: V AT V = V (VAS V) V = V2AS V2 = AS where V2 = I

THEOREM: 8 Let AT  Rn x n is s-symmetric doubly stochastic matrix then AT = VASV. Proof: V AS V = V (VAT V) V =V2AT V2 = AT where V2 = I

THEOREM: 9 Let A, B  Rn x n is s-symmetric doubly stochastic matrix then (A + B) is s-symmetric doubly stochastic matrix. Proof: Let A and B are s-symmetric doubly stochastic matrices if AS = V AT V and BS = VBTV. To prove (A + B) is s-symmetric doubly stochastic matrix we will show that

(A + B)S = V (A + B)T V

Now V (A + B)T V = V (AT + BT ) V = V(AT + BT ) V = (VAT + VBT) V= (VAT V+ VBT V)

= (AS + BS) where V AT V = AS and VBTV = BS

= (A + B)S THEOREM: 10 If A and B are s-symmetric doubly stochastic matrices then AB is also s-symmetric doubly stochastic matrix. Proof: Let A and B are s-symmetric doubly stochastic matrices if AS = V AT V and BS = VBTV. Since AT and BT are also s-symmetric doubly stochastic matrices then AT = VASV and BT= VBSV. To prove A B is s-symmetric doubly stochastic matrix we will show that (AB)S = V(A B )T V Now V (A B)T V= VBTATV = V(VBS V)(VASV)V where AT = KASK and BT = KBSK. = V2 BSV2AS V2 = BS AS where V2 = I = (AB)S www.ijesi.org 13 | Page k - Symmetric Doubly Stochastic, s – Symmetric…

THEOREM: 11 If A is s-symmetric doubly stochastic matrix and V is a permutation matrix with units in the secondary diagonal then VA is also s-symmetric doubly stochastic matrix. Proof: Let A is s-symmetric doubly stochastic matrices if AS = V AT V. Since AT is s-symmetric doubly stochastic matrices then AT = VASV. To prove VA is s-symmetric doubly stochastic matrix we will show that (VA)S= V(VA)T V Now V (VA)T V = V(ATVT )V= V(VASV) V2 = AS V S where V2 = I = (VA)S

RESULT: For A Rn x n is s-symmetric doubly stochastic matrix for the following are holds. [1] AS = V AT V [2] VA is symmetric doubly stochastic matrix. [3] AV is symmetric doubly stochastic matrix.

Example:

A = AT = AS = V =

(i) V AT V = =AS

Similarly VASV = AT

(ii) VA = = = (VA)T

 VA is symmetric doubly stochastic matrix. Similarly VAT is also symmetric doubly stochastic matrix.

(iii) AV = = = (AV)T

 AV is symmetric doubly stochastic matrix. Similarly AT V is also symmetric doubly stochastic matrix.

DEFINITION: 4 A matrix A  Rn x n is said to be s-k-symmetric doubly stochastic matrix if [1] A = KVATVK [2] AT = KVAVK [3] A = VKATKV [4] AT = VKAKV

Where V is a permutation matrix with units in the secondary diagonal and K is a permutation matrix and k = {(1) (2 3)}.

THEOREM: 12 Let A  Rn x n is s-k-symmetric doubly stochastic matrix then [1] AS = KVATVK [2] AT = KVASVK [3] AS = VKATKV [4] AT = VK AS KV

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Proof: (i) KVATVK = K(VATV)K = KAS K where VATV = AS =K(AT)T K = K AT K where AT = A = A = AS where KAT K = A and A = AS (ii) KVASVK = K(VAS V)K = K AT K where VASV = AT = A where K AT K = A = AT where A = AT (iii) VKATKV = V(KATK)V = VAV where KAT K = A = VATV where A = AT = AS where VATV = AS (iv) VKASKV = V(KASK)V = V (K(AT)T K)V = VATV where K(AT )TK = AT = AS where V AT V= AS = (AT )T = AT where A = AT THEOREM: 13 Let A, B  Rn x n is s-k-symmetric doubly stochastic matrix then (A + B ) is s-k-symmetric doubly stochastic matrix. Proof: Let A and B are s-k-symmetric doubly stochastic matrix if A =KV AT VK and B = KVBTVK. To prove (A + B ) is s-k-symmetric doubly stochastic matrix we will show that

(A + B ) = KV (A + B )T VK

Now KV (A + B )T VK = K( V (A + B )T V)K = K (A + B )S K using theorem (9)

= K (A + B )T K = (A + B ) using theorem (3) THEOREM: 14 If A and B are s-k-symmetric doubly stochastic matrix then AB is also s-k-symmetric doubly stochastic matrix.

Proof: Let A and B are s-k-symmetric doubly stochastic matrix if A =KV AT VK and B = KVBTVK. Since AT and BT are also s-k-symmetric doubly stochastic matrices AT = KVAVK and BT = KVBVK. To prove A B is s-k-symmetric doubly stochastic matrix we will show that AB = KV(A B )T VK Now KV(A B )T VK = K(V(A B )T )VK = K(A B )SK using theorem (10) = K(A B )TK = AB using theorem (5)

RESULT: For A Rn x n is s-k-symmetric doubly stochastic matrix the following are equivalent. (i) A = KVATVK (ii) AT = KVAVK (iii) A = VKATKV (iv) AT = VKAKV Example:

A = K = V =

(i) KVATVK = = A

(ii) KVAVK = = AT

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(iii) VKATKV = = A

(iv) VKAKV = = AT

REFERENCES: [1] Ann Lec. Secondary symmetric and skew symmetric secondary orthogonal matrices; i. Period, Math Hungary, 7, 63-70(1976). [2] Hill, R.D, and Waters, S.R: On k-real and k-hermitian matrices; Lin.Alg.Appl.,169,17-29(1992) [3] S.Krishnamoorthy, R.Vijayakumar, On s-normal matrices, Journal of Analysis and Computation, Vol5, No2,(2009) [4] R.Grone, C.R.Johnsn, E.M.Sa,H.Wolkowiez, Normal matrices, Linear Algebra Appl. 87(1987) 213-225. [5] G. Latouche, V. Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modeling, 1st edition. Chapter 2: PH Distributions; ASA SIAM, 1999. [6] Hazewinkel, Michiel, ed. (2001), "Symmetric matrix", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 [7] S.Krishnamoorthy, K. Gunasekaran, N. Mohana, “ Characterization and theorems on doubly stochastic matrices” Antartica Journal of Mathematics, 11(5)(2014).