International Journal of Engineering and Applied Sciences (IJEAS) ISSN: 2394-3661, Volume-4, Issue-3, March 2017
A Modified Minimal Gersgorin Disc Theorem of Matrix Eigenvalue Xi Xu, Guang Zeng, Li Lei, Desheng Ning Abstract— In this paper, a new improvement of the minimum gersgorin disc theorem is given, which uses the improvement of the minimum disc theorem to obtain a new region of eigenvalue distribution. Although the method is not precise, it is very useful in practical applications. Index Terms—Eigenvalue; minimum disc theorem.
Gerschgorin
disc
Lemma 2 (Minimal Gerschgorin disc theorem [2,3]). Suppose
A aij C nn
that
，
X
diag( x1 , x2 ,, xn ), xi 0 ( i N ), let
theorem;
aij x j x ir A z C : z aii ri x A , i N xi j i
I. INTRODUCTION
and
In scientific research, in many problems such as mathematical physics and numerical analysis can be transformed into solving linear system, how to calculate eigenvalues of matrix plays a very important role in the field of matrix analysis for a long time. For some large matrix, it is difficult to find out the exact value of eigenvalues. Hence, we can estimate approximate value of matrix eigenvalue, The disk theorem is a classical method for estimating the eigenvalues of a matrix. In addition, the minimal disc theorem is a more efficient theorem based on the theorem of the matrix. On the basis of previous studies, this paper a new improvement of the minimum gersgorin disc theorem is given, which uses the improvement of the minimum disc theorem to obtain a new region of eigenvalue distribution. Although the method is not precise, it is very useful in practical applications. First we recall some basic Theorem that will be used in this paper. Lemma 1(Gerschgorin disc theorem [1]). Suppose that
r A ir A, R A r A. x
x
x
iN
X 0
Then A is minimal Gerschgorin disc region of proof. Suppose that R
a11 a A 21 a n1
A.
a1n a2 n , ann
a12 a22 an 2
we get
a11 a21 x1 1 X AX = x 2 an1 x1 x n
a1n xn x1 a x a22 2 n n x2 . a n 2 x2 ann xn 1 Due to definition of eigenvalue X AX Y Y , | yi | max(| y j |), j 1, 2,, n. Thus
A aij C nn ，n N and is an eigenvalue of A ，let
A is spectrum of A , then A A i A, iN
where
i A z C : z aii ri A aij i N . j i A named Gerschgorin region of A, i A named
j
aij x j xi
a12 x2 x1
y j y j , i 1, 2, , n,
then
aii yi
number i Gerschgorin disc.
aij x j
j i
aii yi j i
yj, xi aij x j yj , xi
we obtain
Xi Xu, School of Sciences, East China University of Technology, Nanchang, 330013, P.R. China. Guang Zeng, Corresponding author, School of Sciences, East China University of Technology, Nanchang, 330013, P.R. China. Li Lei, School of Sciences, East China University of Technology, Nanchang, 330013, P.R. China. Desheng Ning, School of Sciences, East China University of Technology, Nanchang, 330013, P.R. China..
aii j i
57
aij x j xi
.
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A Modified Minimal Gersgorin Disc Theorem of Matrix Eigenvalue
' of X 1 AX , ' A , then A , R we all the arbitrary of X , then A . x
x
For
a x a r x A pq q pp p xp a x a r x A qp p qq q xq
The proof of Lemma 2 is completed. II. MAIN RESULTS In this section, based on the minimal Gerschgorin disc region of matrix, we state our main results.
Theorem 3. Suppose that A aij C
n n
a pq aqp
, n N , then
Due to the arbitrary of
A D R A R A,
for
where
X diagx1 , x2 ,, xn , then all eigenvalues for A
In order to proof that the theorem is more effective than the original minimum disc theorem, so we need to proof
D R A R A, R for z D A and X 0 ， i, j N , exit a x z a r x A ij j ii i xi a x z a r x A ji i jj j xj
D x A Dix A, iN
D A D x A,
aij a ji
R
X 0
easy
to
According to Lemma 2, if
known
z a jj rjx A, j i,
that
X 1 AX . Assuming that y is called an eigenvector T n of A associated with , y ( y1 , y2 ,, yn ) R , due to definition of eigenvalue,
X
let
1
AX y y
and
maximum value of each component of eigenvector y,
respectively, i.e. min y p , yq yi , i p, q. Then
a pp
z aii ri x A
aij x j
z a jj rjx A
a ji xi
a pq xq a pq xq yq , xp xp y p
So
aqp x p aqp x p y p xq xq yq
a ji xi xj
,
(2.6)
,
(2.7)
aij a ji .
(2.2) in the same way,
aqq rqx A
xj
xi
a x z a r x A ij j ii i xi a x z a r x A ji i jj j xj
same as
aij x j
According to (2.6), (2.7), we can obtain
a pq xq a x y pq q y , y p rpx A p q xp xp
a pp rpx x
xi
(2.1)
| y p |, | yq | be the first maximum value and the second
(2.5)
X 0 , z R A, then z aii ri x A,
X diagx1, x2 ,, xn xi 0 i N . we
and (2.4), we could obtain,
must be seated the union
j i
For A,
, y p , yq
D R A compose with then Vijx A , in short A D R A.
a x z C : aii ri x A ij j xi a x a r x A ji i Vijx A jj j xj aij a ji x Di A Vijx A,
Proof.
(2.4)
we
get
a
D R A R A.
contradiction
with
the
know,
then
The proof of Theorem 3 is completed.
(2.3)
Corollary 4 If A A aij C T
According to (2.2), (2.3), we can obtain
n n
, n N , then
A D A A, R
58
R
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International Journal of Engineering and Applied Sciences (IJEAS) ISSN: 2394-3661, Volume-4, Issue-3, March 2017 where
Vijx A z C : z aii z a jj six A s xj A
Di A Vijx A, x
j i
D A Dix A, x
iN
D A D x A, R
X 0
six A j i
a ji xi xj
s xj A ,
i j
aij x j xi
X diagx1, x2 ,, xn xi 0i N . III. CONCLUSION
In many fields such as numerical analysis and mathematical physics, a number of problems come down to the study of linear system, how to calculate eigenvalues of matrix plays a very important role in the field of matrix analysis for a long time. For some large matrix, it is difficult to find out the exact value of eigenvalues. On the basis of previous studies, this paper a new improvement of the minimum gersgorin disc theorem is given, which uses the improvement of the minimum disc theorem to obtain a new region of eigenvalue distribution. Although the method is not precise, it is very useful in practical applications. In addition, in the process of application, the main conclusions of the diagonal matrix X is particularly important, we generally choose the diagonal element between 0 and 1, the method is more effective. The selection of X technology is worth our further exploration. ACKNOWLEDGMENT This research was supported in part by the National Natural Science Foundation of China(11661005, 11301070), the Natural Science Foundation of Jiangxi Province of China(20151BAB211012), the Science Foundation of Education Committee of Jiangxi for Young Scholar (GJJ150576). REFERENCES [1]
[2]
[3] [4] [5]
Junjing Chen, Xifan Zhang, Feng Shan. Study on Estimation and Location of Matrix Eigenvalue. Shenyang Journal of Shenyang institute of aeronautical engineering, 1998, 15(4):41-45. Ruifang Li, Chuanlong Wang. The Technology and Improvement of Minimal Disc Theorem. Journal of Taiyuan Normal University(Natural Science Edition), 2013, 12(2):41-43. Ruifang Li. The Minimal Gersgorin Disk Theorem and its GeneralizaiZation[D]. Taiyuan: Shanxi University, 2014. Kostic V., Cvetkovic L. J., Varga R. S. Gersgorin-type localizations theorem. Adv. Comput. Math., 2011, 35:271-280. Cvetkovic L. J., Kostic V., Between gersgorin and minimal gersgorin sets. Comput. Appl. Math., 2006, 196(2):452-458.
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