The document discusses methods for estimating the eigenvalues of quaternion doubly stochastic matrices. It presents several theorems: 1) The eigenvalues of a quaternion doubly stochastic matrix A are located within Gerschgorin balls centered at the minimum diagonal entry of A with radius 1 less than the minimum diagonal entry. 2) The eigenvalues of A are located within balls centered at the average trace of A with radius related to the maximum entry of A. 3) The eigenvalues of the tensor product of two quaternion doubly stochastic matrices A and B are located within an oval region.

1. International Journal of Trend in Scientific Research and Development (IJTSRD)
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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
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ABSTRACT
The purpose of this paper is to locate andestimatethe eigenvaluesofquaternion
doubly stochastic matrices. We present several estimation theorems about the
eigen values of quaternion doublystochasticmatrices. Meanwhile,weobtainthe
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
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 combinationmatrixtheory, 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 quaterniondoublystochasticmatrixsection2
introduces the concept of quaternion doubly stochastic
matrix and generalized quaterniondoublystochasticmatrix.
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
T
A are quaternion row stochastic matrices,
A is quaternion double stochastic matrix; Row stochastic
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 quaterniondoublestochasticmatrix are
called the first generalized quaternion stochastic matrix,
denoted by IS (n) .
Definition 3
The second generalized quaternion row stochastic matrix,
the second generalized quaternion column stochasticmatrix
and the second generalized quaternion double stochastic
matrix are called the second generalized quaternion
stochastic matrix, denoted by IIS (n) .
Definition 4
The third generalized quaternion row stochastic matrix, the
third generalized quaternion column stochastic matrix and
the third generalized quaternion double stochastic matrix
IJTSRD22860

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are called the third generalized quaternion stochasticmatrix,
denoted by IIIS (n) .
IS (n) , IIS (n) , IIIS (n) are called generalized stochastic
matrices obviouslyfor S(n) , IS (n)
, IIS (n) and IIIS (n) we
have the following simple conclusions
1. I IIIS(n) S (n) S (n)⊂ ⊂
2. II IIIS(n) S (n) S (n)⊂ ⊂
3. II IIIS(n) S (n) S (n)⊂ ⊂
Theorem 1
Suppose cd n nA (a ) ×= is a quaternion doubly stochastic
matrix and { }ccm min a ,c,d 1,2,...,n= = then
{ }(A) G(A) Z: Z m 1 mλ ⊂ = − ≤ − . Where (A)λ is
denoted the whole eigen values of matrix A , G(A)is
gerschgorin balls of matrix A .
Proof:
Since λ is an arbitrary eigen value of matrix
cd n nA (a ) ×= and
T
1 2 nX (x ,x ,...,x )= ∈ n 1
H ×
is the
corresponding column eigen vector, let
c
c
c
x
y
t
= where ct (c 1,2,...,n)= ,
c cy max y (c 1,2,...,n)= = and from Ax x= λ
n
c c cd d d
d 1
y t a t y
=
λ = ∑
n
e e cd d d
d 1
d e
y t a t y
=
≠
λ = ∑
n
e e ee e e cd d d
d 1
d e
y t a t y a t y
=
≠
λ = + ∑
Multiply right each item of the above equation with ey ∗
n
e e e ee e e e cd d d e
d 1
d e
y t y a t y y a t y y∗ ∗ ∗
=
≠
λ = + ∑
n
e e e ee e e e cd d d e
d 1
d e
t y y a t y y a t y y∗ ∗ ∗
=
≠
λ = + ∑
n
e e e ee e e e cd d d e
d 1
d e
t y y a t y y a t y y∗ ∗ ∗
=
≠
λ = + ∑
n
d ed d e
e ee e 2
d 1 e
d e
t a y y
( t a t )
y
∗
=
≠
λ − = ∑
By triangular equality
n
e ee e d cd
d 1
d e
t a t t a
=
≠
λ − ≤ ∑
n
e ee e cd
d 1
d e
t a Q a
=
≠
λ − ≤ = ∑
ee1 a= −
Therefore,
ee ee ee ee ee eee a a e a a e 1 a a e 1 eλ − = λ − + − ≤ λ − + − ≤ − + − = −
Since λ is an arbitrary eigen value of quaternion doubly
stochastic matrix cd n nA (a ) ×= , then
{ }(A) G(A) Z: Z m 1 mλ ⊂ = − ≤ − .
So the eigen values of A are located in the gerschgorin balls
whose center { }ccm min a ,c,d 1,2,...,n= = and radius
1 m− .
Hence proved.
Theorem 2
Suppose cd n nA (a ) ×= is a quaternion doubly stochastic
matrix and
{ }cd cd
cd
M max a ,c,d 1,2,...,n= = then
2n n
cd
d 1 c 1
Tra(A) n 1 (Tra(A))
(A) G(A) Z: Z ( M )
n n n= =
 − 
λ ⊂ = − ≤ − 
  
∑∑
Where (A)λ is denoted the whole eigen values of matrix
A . G(A)is denoted balls whose center is
Tra(A)
n
and
radius is
2n n
cd
d 1 c 1
Tra(A) n 1 (Tra(A))
(A) G(A) Z: Z ( M )
n n n= =
 − 
λ ⊂ = − ≤ − 
  
∑∑
Proof:
From paper (yixiGu, 1994) and for arbitrary matrix A
( ) ( )( )
2
2
F
Tra ATra(A) n 1
A
n n n
−
λ − ≤ − and because
cd n nA (a ) H(n)×= ∈ ,
n n
2
cd cdF
c 1 d 1
A M M
= =
≤ ∑ ∑ .Sowehave
( )( )
2
n n
cd
c 1 d 1
Tra ATra(A) n 1
(A) G(A) Z: Z M
n n n= =
 
− 
λ ⊂ = − ≤ − 
  
∑∑
Similarly we get,
( )( )
2
Tra ATra(A) n 1
(A) G(A) Z: Z 1
n n n
  −  λ ⊂ = − ≤ − 
    
This completes the proof of the theorem.
Theorem 3
Suppose cd n nA (a ) ×= and ( )cd m m
B b ×
= are quaternion
doubly stochastic matrices, { }1 ccm min a ,c 1,2,..,n= =
and radius is 11 m− , and Gerschgorine balls whose centeris

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{ }2 ddm min a ,d 1,2,...,m= = and radius is
{ }
n
cc 1 cc
c,d 1
(A) G(A) Z; Z a (n 1)M (1 a )
=
λ ⊂ = − ≤ − −U .
Proof:
Let { }1 2 n(A) , ,...,λ = λ λ λ and { }1 2 n(B) , ,...,λ = µ µ µ ,
{ }1 1(A) Z: Z m 1 mλ ⊂ − ≤ − ,
{ }2 2(A) Z: Z m 1 mλ ⊂ − ≤ − and since
( ) [ ]c dA Bλ ⊗ = λ µ c 1,2,...,n= , d 1,2,...,n= .
Therefore, the eigen values of tensor product for matrix A
and Matrix Bare located in the oval region ( )G A B⊗ .
Hence proved.
Theorem 4
Suppose cd n nA (a ) ×= is quaternion doubly stochastic
matrix and { }1 cd
cd
M max a ,c,d 1,2,...,n= = ,
{ }
n
cc 1 cc
c,d 1
(A) G(A) Z; Z a (n 1)M (1 a )
=
λ ⊂ = − ≤ − −U
where (A)λ is denoted thewholeeigen valuesof matrix A ,
G(A)is denoted thegeneralized Greschgorin ballsofmatrix
A .
Proof:
Because λ is an arbitrary eigen value of matrix
cd n nA (a ) ×= and ( )
T n 1
1 2 nX x ,x ,...,x H ×
= ∈ is the
corresponding column eigenvector for Ax x= λ , we set
So,
n
ed d xe
d 1
d e
a x
=
≠
= λ∑
( )
n
ee e ed d
d 1
d e
a x a x
=
≠
λ − = ∑
From schwarz inequality n triagonal inequality, we have the
following result.
2
ed d e
2d e d
ee ed2
d e d e ee
a x x
x
a a
xx
∗
≠
≠ ≠
λ − = ≤
∑
∑ ∑
2 2
ed d e
2 2d e d e
ee ed ed e2
d e d e d ee ee
a x x
x x
a a (n 1) a (n 1) R
x xx
∗
∗
≠
≠ ≠ ≠
λ − = ≤ ≤ − = −
∑
∑ ∑ ∑
Where
2
e ed
d e
R a ,e 1,2,...,n
≠
= =∑ and since
2
e ed e ed e ee
d e d e
R a M a M (1 a ),
≠ ≠
= ≤ = −∑ ∑
e 1,2,...,n= , ee e eea (n 1)M (1 a )λ − ≤ − −
holdsbecause λ is an arbitrary eigenvalue of a matrix A.
{ }
n
cc 1 cc
c,d 1
(A) G(A) Z; Z a (n 1)M (1 a )
=
λ ⊂ = − ≤ − −U
The theorem is proven.
Eigen value estimate for generalize quaternion doubly
stochastic matrix.
Theorem 5: (yuanlu, 2010)
Suppose cd n n IIIA (a ) S (n)×= ∈ , cca and dda are the most
small diagonal elements in A , then
{ }cc dd cc dd(A) G(A) Z: Z a Z a (S a )(S a )λ ⊂ = − − ≤ − −
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 IIIA (a ) S (n)×= ∈ and
cd m m IIIB (b ) S (m)×= ∈ are quaternion doubly stochastic
matrices
{ }cc dd cc dd(A B) G(A B) Z: Z a Z a (S a )(S a )λ ⊗ ⊂ ⊗ = − − ≤ − −
{ }cc dd cc ddZ: Z b Z b (S b )(S b )− − ≤ − −
Where (A B)λ ⊗ is denoted the whole eigen values of
tensor product for matrix A and matrix B , G(A B)⊗ is
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 leavenforreaders
Theorem 7:
Suppose cd n n IIIA (a ) S (n)×= ∈ and
{ }ccm min a ,c 1,2,...,n= = , then
{ }(A) G(A) Z: Z m S eλ ⊂ = − ≤ +
Where (A)λ is denoted the wholeeigen values of matrix
A , G(A) is the balls whose center is
{ }ccm min a ,c 1,2,...,n= = and radius S e+
Proof:
From Gerschgorin balls theorem, we have
n
ee e ed ee
d 1
a Q a S a
=
λ − ≤ = = −∑
Therefore,
ee ee ee ee ee eee a a e a a e S a a e S eλ − = λ − + − ≤ λ − + − ≤ − + + = +
Because λ is an arbitrary eigen value of matrix
cd n nA (a ) ×=
{ }(A) G(A) Z: Z e 1 eλ ⊂ = − ≤ +
So the eigen values of matrix A are locatedinthediscwhose
center is { }ccm min a ,c 1,2,...,n= = and radius is teS .

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Theorem 8 :
Suppose cd n n IIIA (a ) S (n)×= ∈ , cd m m IIIB (b ) S (m)×= ∈
and { }1 ccm min a ,c 1,2,...,n= = and
n n
1 cd cd
c 1 d 1
m max a . a ,c,d 1,2,...,n
= =
 ′ = = 
 
∑ ∑
{ }2 ddm min b ,d 1,2,...,m= =
,
n n
2 cd cd
c 1 d 1
m max b . b ,c,d 1,2,...,n
′
= =
 ′ = = 
 
∑ ∑
Then
{ } { }cc 2 cc dd 2 dd(A B) G(A B) Z: Z a (n 1)m (1 a ) Z: Z a (n 1)m (1 a )′λ ⊗ ⊂ ⊗ = − ≤ − − • − ≤ − −
Where (A B)λ ⊗ is denoted the whole eigen values of
tensor for matrix A and matrix B , G(A B)⊗ is the oval
region of the product for elements of balls whose center is
{ }1 ccm min a ,c,d 1,2,...,n= = and radius 1S m+ and
balls whose center is { }2 ddm min b ,d 1,2,...,m= = and
radius 2S m+ .
Proof:
{ } { }min min cc 1 cc dd 2 dd(A B) G (A B) Z: z a (n 1)m (1 a ) Z: z a (n 1)m (1 a )λ ⊗ ⊂ ⊗ = − ≤ − − • − ≤ − −
2n n
max max 1 2
c 1 d 1
Tra(A B) n 1 (Tra(A B))
(A B) G (A B) Z: z m m
n n n= =
 ⊗ − ⊗ ′ ′λ ⊗ ⊂ ⊗ = − ≤ • − 
  
∑∑
Therefore, the maximum and minimum eigenvalues of
tensor product for Matrix A and B. Hence proved.
Theorem
Suppose cd n nA (a ) S(n)×= ∈ , cca and dda ar the most
small module diagonal cross elements in A , then
( )( ){ }cc dd cc dd(A) G(A) Z: Z a Z a S a S aλ ⊂ = − − ≤ + +
Where (A)λ is denotedthe wholeeigenvaluesof matrix A ,
G(A) is denoted cassini oval region of matrix A .
Theorem
Suppose cd n n IIIA (a ) S (n)×= ∈ and
cd m m IIIB (b ) S (m)×= ∈ are quaternion doubly stochastic
matrices, then
{ }cc dd cc dd(A B) G(A B) Z: Z a Z a (S a )(S a )λ ⊗ ⊂ ⊗ = − − ≤ + +
{ }cc dd cc ddZ: Z b Z b (S b )(S b )− − ≤ + +
Where (A B)λ ⊗ is denoted the whole eigen values of
tensor product for matrix A and matrix B , G(A B)⊗ is
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 stochasticmatrices.Finally,weobtain the
eigenvalues using tensor product of two quaternion doubly
stochastic matrices.
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