The Estimation for the Eigenvalue of Quaternion Doubly Stochastic Matrices using Gerschgorin Balls

1. International Journal of Trend in Scientific Research and Development (IJTSRD) Volume: 3 | Issue: 3 | Mar-Apr 2019 Available Online: www.ijtsrd.com e-ISSN: 2456 - 6470 The Estimation for the Eigenvalue of Quaternion Doubly Stochastic Matrices using Gerschgorin Balls Dr. Gunasekaran K1, Mrs. Seethadevi R2 1Head and Associative Professor, 2PhD Scholar 1,2Department of Mathematics, Government Arts College (Autonomous), Kumbakonam, Tamil Nadu, India How to cite this paper: Dr. Gunasekaran K | Mrs. Seethadevi R "The Estimation for the Eigenvalue of Quaternion Doubly Stochastic Matrices using Gerschgorin Balls" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456- 6470, Volume-3 | Issue-3, April 2019, pp.667-670, URL: http://www.ijtsrd.co m/papers/ijtsrd228 60.pdf Copyright © 2019 by author(s) and International Journal of Trend in Scientific Research and Development Journal. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0) (http://creativecommons.org/licenses/ by/4.0) INTRODUCTION In the past two decades due to study on matrix theory and some engineering background problems, many scholars dedicated to special matrix, and obtained some important and valuable results. Hing zhu-2007 yigeng Huang-1994). But in combination matrix theory, combinatorics, probability theorey. Mathematical economics and reliability theory etc., area there is a special class of non-negative quaternion doubly stochastic matrix, which in recent years becomes concerned. The article discuss location, distribution and estimate of the eigen value for quaternion doubly stochastic matrix section 2 introduces the concept of quaternion doubly stochastic matrix and generalized quaternion doubly stochastic matrix. Section 3 gives a few estimation theorems of quaternion doubly stochastic matrix eigen value, also the eigen value distribution for tensor Product of two quaternion doubly stochastic matrices is obtained. In section 4 we discuss the eigen value distribution for generalized quaternion doubly stochastic matrices, (3) eigen value estimate of quaternion doubly stochastic matrix. Definition 1 If both A and A are quaternion row stochastic matrices, A is quaternion double stochastic matrix; Row stochastic ABSTRACT The purpose of this paper is to locate and estimate the eigen values of quaternion doubly stochastic matrices. We present several estimation theorems about the eigen values of quaternion doubly stochastic matrices. Mean while, we obtain the distribution theorem for the eigen values of tensor products of two quaternion doubly stochastic matrices. We will conclude the paper with the distribution for the eigen values of generalized quaternion doubly stochastic matrices. KEYWORDS: Quaternion doubly stochastic matrices, Eigen values and tensor product, gerschgorin balls. AMS Classification: 15A03, 15A09, 15A24, 15B27 IJTSRD22860 matrix, column stochastic matrix and quaternion double stochastic matrix are called stochastic matrix, denoted by S(n) . Definition 2 The first generalized quaternion row stochastic matrix, the first generalized quaternion column stochastic matrix and the first generalized quaternion double stochastic matrix are called the first generalized quaternion stochastic matrix, denoted by Definition 3 The second generalized quaternion row stochastic matrix, the second generalized quaternion column stochastic matrix and the second generalized quaternion double stochastic matrix are called the second generalized quaternion stochastic matrix, denoted by Definition 4 The third generalized quaternion row stochastic matrix, the third generalized quaternion column stochastic matrix and the third generalized quaternion double stochastic matrix IS (n) . S (n) . II T @ IJTSRD | Unique Paper ID – IJTSRD22860 | Volume – 3 | Issue – 3 | Mar-Apr 2019 Page: 667

2. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 are called the third generalized quaternion stochastic matrix, denoted by III S (n) . IS (n) , matrices obviously for S(n) , n =∑ λ − ≤ t a Q a e ee e cd = d 1 d e ≠ S (n) are called generalized stochastic IS (n), have the following simple conclusions 1. I III S(n) S (n) S (n) ⊂ ⊂ 2. II III S(n) S (n) S (n) ⊂ ⊂ 3. II III S(n) S (n) S (n) ⊂ ⊂ Theorem 1 Suppose cd n n A (a )× = is a quaternion doubly stochastic m min a = { (A) G(A) Z: Z m denoted the whole eigen values of matrixA , G(A)is gerschgorin balls of matrixA . Proof: Since λ is an arbitrary eigen value of matrix A (a )× = and 1 X (x ,x ,...,x ) = corresponding column x y t y max y (c 1,2,...,n) = = S (n) , II III 1 a = − S (n) and S (n) we ee II III Therefore, e λ− Since λ is an arbitrary eigen value of quaternion doubly stochastic matrix A { (A) G(A) Z: Z m So the eigen values of A are located in the gerschgorin balls whose center { cc m min a ,c,d = 1 m − . Hence proved. Theorem 2 Suppose cd n n A (a )× = is a quaternion doubly stochastic matrix and { cd cd cd   λ ⊂ = − ≤    Where (A) λ is denoted the whole eigen values of matrix A . G(A)is denoted balls whose center is Tra(A) e 1 e − = − = λ− + − ≤ λ− + − 1 a ≤ − + a a e a a e a ee ee ee ee ee ee = cd n n (a )× 1 m ≤ − , then } λ ⊂ = − . { ≤ − } (A) } = = ,c,d } 1,2,...,n 1,2,...,n matrix and then and radius cc λ λ ⊂ = − 1 m . Where is } = = M max a ,c,d 1,2,...,n then ∈ n 1 × is the vector, T H 2 n cd n n      − 2 Tra(A) n n 1 n (Tra(A)) n n n eigen let ∑∑ − (A) G(A) Z: Z ( M ) cd d 1 c 1 = = = = c ct (c 1,2,...,n) where , c c = λ and from Ax =∑ x and c c n n radius is y t λ cd d a t y c c d           − 2 Tra(A) n n 1 n (Tra(A)) n n n ∑∑ d 1 = n λ ⊂ = − ≤ − (A) G(A) Z: Z ( M ) cd d 1 c 1 = = =∑ λ y t cd d a t y Proof: From paper (yixiGu, 1994) and for arbitrary matrix A ( n n e e d = d 1 d e ≠ n +∑ λ = y t ee e a t y cd d a t y ( ) ( ) n n ∑ 2 ) Tra A e e e d n 1 − Tra(A) 2 = λ− ≤ − d 1 d e ≠ A and because F Multiply right each item of the above equation with n ey∗ ≤∑ 2 = ∈ A M M A cd n n (a ) H(n) , . So we have × cd cd F n c 1 = d 1 = +∑ ∗ ∗ ∗ λ = y t y ee e a t y y cd d a t y y           ( ) ( ) n 2 e e e e e d e Tra A − Tra(A) n n 1 n n n ∑∑ λ ⊂ = − ≤ − (A) G(A) Z: Z M = d 1 d e ≠ n cd c 1 d 1 = = +∑ Similarly we get, ∗ ∗ ∗ λ = t y y ee e a t y y cd d a t y y e e e e e d e           ( )         ( ) n 2 = Tra A d 1 d e ≠ n − Tra(A) n n 1 n λ ⊂ = − ≤ − (A) G(A) Z: Z 1 +∑ ∗ ∗ ∗ λ = t y y ee e a t y y cd d a t y y e e e e e d e = d 1 d e ≠ This completes the proof of the theorem. Theorem 3 Suppose cd n n A (a )× = ∗ t a y y n =∑ ( t λ − ee e a t ) d ed y d 2 e e ( min a ) { = = = d 1 d e ≠ B b and are quaternion e cd m m × } By triangular equality = m ,c 1,2,..,n doubly stochastic matrices, 1 cc 1 m − n and radius is , and Gerschgorine balls whose center is ≤∑ λ − 1 t ee e a t t a e d cd = d 1 d e ≠ @ IJTSRD | Unique Paper ID - IJTSRD22860 | Volume – 3 | Issue – 3 | Mar-Apr 2019 Page: 668

3. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 { 2 dd min a ,d 1,2,...,m { c,d 1 = } { } n = = m and radius is } c,d 1 U The theorem is proven. λ ⊂ = − ≤ (n 1)M (1 a ) − − (A) G(A) Z; Z a cc 1 cc n = U λ ⊂ = − ≤ (n 1)M (1 a ) − − (A) G(A) Z; Z a . cc 1 cc Eigen value estimate for generalize quaternion doubly stochastic matrix. Theorem 5: (yuanlu, 2010) Suppose cd n n III A (a ) S (n) × = ∈ small diagonal elements in A , then { cc (A) G(A) Z: Z a Where (A) λ is denoted the whole eigen values of matrix A,G(A) is denoted cassini oval region of matrix A . Theorem 6: (yancheng, 2010) Suppose cd n n A (a ) = B (b ) S (m) × = ∈ are quaternion doubly stochastic matrices { (A B) G(A B) Z: Z a { cc dd Z: Z b Z b Where (A B) λ ⊗ is denoted the whole eigen values of tensor product for matrix A and matrix B , G(A the oval region of the product for cassini oval region elements of matrix A and cassini oval region elements of matrix B Proof: This proof is same to theorem(3) which is leaven for readers Theorem 7: Suppose cd n n A (a ) = { cc m min a ,c { (A) G(A) Z: Z m Where (A) λ is denoted the wholeeigen values of matrix A ,G(A) is the balls whose center is { cc m min a ,c 1,2,...,n Proof: From Gerschgorin balls theorem, we have n a Q λ− ≤ = ∑ Proof: Let { } { } λ = λ λ λ λ = µ µ µ (A) , ,..., (B) , ,..., and } 1 , 1 2 n 1 2 n a a , and are the most { { B λ ⊂ − 1 m ≤ − cc dd (A) Z: Z m , 1 } λ λ Therefore, the eigen values of tensor product for matrix A and Matrix Bare located in the oval region ( Hence proved. Theorem 4 Suppose cd n n A (a )× = is quaternion doubly stochastic M max a = { c,d 1 = where (A) λ is denoted the whole eigen values of matrix A , G(A)is denoted the generalized Greschgorin balls of matrix A . Proof: Because λ is an arbitrary eigen value of matrix A (a )× = and ( 1 X x ,x ,...,x = corresponding column eigenvector for Ax n a x = ≠ ( ee e a x λ− ⊂ ⊗ − 1 m ≤ − = (A) ( A Z: Z m ) [ = λ µ and since 2 ] 2 } λ ⊂ = − Z a − ≤ (S a )(S a ) − − , d 1,2,...,n = c 1,2,...,n dd cc dd . c d ) ⊗ G A B . ∈ S (n) and × III cd m m III { } = ,c,d 1,2,...,n matrix and , } 1 cd λ ⊗ ⊂ ⊗ = − Z a − ≤ (S a )(S a ) − − cd cc dd cc dd } n U λ ⊂ = − ≤ (n 1)M (1 a ) − − (A) G(A) Z; Z a } cc 1 cc − − ≤ (S b )(S b ) − − cc dd ⊗ B) is ) T n 1 × ∈ = λ , we set H is the 2 n cd n n x ∑ = λ So, ∈ S (n) and ed d xe × 1,2,...,n III d 1 d e } ≤ + = = , then n =∑ ) } a x λ ⊂ = − S e ed d = d 1 d e ≠ From schwarz inequality n triagonal inequality, we have the following result. a x x a x } ∑ ∗ + = = and radius S e 2 ed d e x x ∑ ∑ 2 λ− = ≤ d e ≠ a d ee ed 2 d e ≠ d e ≠ e e ∑ ∗ a x x 2 2 ∗ ed d e x x x x ∑ ∑ ∑ 2 2 λ− = ≤ ≤ (n 1) − = (n 1)R − d e ≠ a a a = − d e a S a ee ed ed e 2 x ee e ed ee d e ≠ d e ≠ d e ≠ e e e d 1 = ∑ 2 = ,e 1,2,...,n = Therefore, e λ− = λ− Because λ is an arbitrary eigen value of matrix A (a )× = { (A) G(A) So the eigen values of matrix A are located in the disc whose center is { cc m min a ,c 1,2,...,n R a Where and since e ed + = + + − ≤ λ− + − ≤ − + a a e a a e S a a e S e d e ≠ ee ee ee ee ee ee ∑ ∑ 2 = ≤ = M (1 a ), − R a M a e ed e ed e ee cd n n d e ≠ d e ≠ ≤ } λ ⊂ = − 1 e ≤ + Z: Z e λ− (n 1)M (1 a ) − − = a e 1,2,...,n , ee e ee holdsbecause λ is an arbitrary eigenvalue of a matrix A. } = = S . and radius is te @ IJTSRD | Unique Paper ID - IJTSRD22860 | Volume – 3 | Issue – 3 | Mar-Apr 2019 Page: 669

4. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 Theorem 8 : Suppose Theorem Suppose B (b ) = matrices, then (A λ ⊗ { Z: Z b Where λ tensor product for matrix A and matrix B, G(A the oval region of the product for cassini oval region elements of matrix A and cassini oval region elements of matrix B. Conclusion In this paper is to locate and estimate the eigenvalues of quaternion doubly stochastic matrices then we present a several estimation theorems about the eigen values of quaternion doubly stochastic matrices. Finally, we obtain the eigenvalues using tensor product of two quaternion doubly stochastic matrices. References: [1]Brauer.A (1964). On nonnegative matrices, Recent advances in Matrix Theory. Wiles Press. 3-38 = ∈ = ∈ = ∈ A cd n n (a ) m = S (n) { cc ∑ { dd B cd m m (b ) 1,2,...,n S (m) A cd n n (a ) are quaternion doubly stochastic S (n) , = and × × × III III III } ∈ S (m) min a ,c and and × cd m m III 1    min b    n n { } ∑ ′= = ⊂ ⊗ = − Z a − + ≤ + + B) G(A − ⊗ B) Z: Z a − (S a )(S + a } ) m max a . a ,c,d 1,2,...,n cc (S dd b )(S cc dd 1 cd cd c 1 = d 1 = ≤ Z b b ) } cc dd cc dd = = m ,d 1,2,...,m (A B) 2 is denoted the whole eigen values of , ⊗       B) ′ n n is ∑ ∑ ′= = m max b . b ,c,d 1,2,...,n 2 cd cd c 1 = d 1 = Then } { } { B) ′ λ ⊗ ⊂ ⊗ = − ≤ (n 1)m (1 a ) − − • − ≤ (n 1)m (1 a ) − − (A B) G(A λ B) ⊗ Z: Z a Z: Z a cc 2 cc dd 2 dd Where (A tensor for matrix A and matrix B , G(A region of the product for elements of balls whose center is { 1 cc m min a ,c,d 1,2,...,n m min b = S m + . Proof: {  ⊗  λ ⊗ ⊂ ⊗ = −    Therefore, the maximum and minimum eigenvalues of tensor product for Matrix A and B. Hence proved. Theorem Suppose cd n n A (a ) S(n) × = ∈ small module diagonal cross elements in A , then { Where (A) λ is denoted the whole eigen values of matrix A , G(A) is denoted cassini oval region of matrix A . is denoted the whole eigen values of ⊗ B) is the oval } = = S m + and radius and 1 { } = ,d 1,2,...,m balls whose center is and 2 dd radius 2 the characteristic roots } { } λ ⊗ ⊂ ⊗ = − ≤ (n 1)m (1 a ) − − • − ≤ (n 1)m (1 a ) − − (A B) G (A B) Z: z a Z: z a min min cc 1 cc dd 2 dd      − ⊗ 2 Tra(A B) n 1 n (Tra(A B)) n n [2]Gu, Yixi. (1994). The distribution of eigenvalues of matrix. Acta mathematics Applicatae Sinica, (04), 501- 511. ∑∑ ′ ′ ≤ • − (A B) G (A B) Z: z m m max max 1 2 n n c 1 d 1 = = [3]Guo, Zhong. (1987). The existence of real double stochastic wit hpositive eigenvalue. Mathem,atics in practice and theory, (01). a a , and ar the most cc dd [4]Lin, Lvuduan. (1984). Estimates of determinant and eigenvalue for two types matrices of row module equaling. Math ematics in practice and theory, (04). } ( )( ) λ ⊂ = − Z a − ≤ + + (A) G(A) Z: Z a S a S a cc dd cc dd [5]Shen, Guangxing. (1991). The Distribution of Eigenvalue of Generalized Stochastic Matrix. Chinese Journal of Engineering Mathematics, (03), 111-115. [6]Tong, Wenting. (1997). Eigenvalue distribution on several types of matrices. Acta mathematics Sinica, (04). @ IJTSRD | Unique Paper ID - IJTSRD22860 | Volume – 3 | Issue – 3 | Mar-Apr 2019 Page: 670