Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2011, Article ID 245767, 6 pages
doi:10.1155/2011/245767
Research Article
A Note on Kantorovich Inequality for
Hermite Matrices
Zhibing Liu,1, 2 Kanmin Wang,2 and Chengfeng Xu2
1
2
School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
Department of Mathematics, Jiujiang University, Jiujiang 332005, China
Correspondence should be addressed to Zhibing Liu, liuzhibingjju@126.com
Received 10 November 2010; Accepted 19 February 2011
Academic Editor: Sin E. Takahasi
Copyright q 2011 Zhibing Liu et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
A new Kantorovich type inequality for Hermite matrices is proposed in this paper. It holds for the
invertible Hermite matrices and provides refinements of the classical results. Elementary methods
suffice to prove the inequality.
1. Introduction
We first state the well-known Kantorovich inequality for a positive definite Hermite matrix
see 1, 2, let A ∈ Mn be a positive definite Hermite matrix with real eigenvalues λ1 ≤ λ2 ≤
· · · ≤ λn . Then
1 ≤ x∗ Axx∗ A−1 x ≤
λ1 λn 2
,
4λ1 λn
1.1
for any x ∈ Cn , x 1, where A∗ denotes the conjugate transpose of the matrix A. An
equivalent form of this result is incorporated in
0 ≤ x∗ Axx∗ A−1 x − 1 ≤
λn − λ1 2
,
4λ1 λn
1.2
for any x ∈ Cn , x 1.
This famous inequality plays an important role in statistics and numerical analysis, for
example, in discussions of converging rates and error bounds of solving systems of equations
see 2–4. Motivated by interests in applied mathematics outlined above, we establish in this
2
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paper a new Kantorovich type inequality, the classical Kantorovich inequality is modified to
apply not only to positive definite but also to all invertible Hermitian matrices. An elementary
proof of this result is also presented.
In the next section, we will state the main theorem and its proof. Before starting, we
quickly review some basic definitions and notations. Let A ∈ Mn be an invertible Hermite
matrix with real eigenvalues λ1 ≤ λ2 ≤ · · · ≤ λn , and the corresponding orthonormal
eigenvectors ϕ1 , ϕ2 , . . . , ϕn with ϕi 1 i 1, 2, . . . , n, where ϕi denotes 2-norm of the
vector of Cn .
For A, we define the following transform
CA, x x∗ λn I − AA − λ1 Ix.
1.3
1
1
A−1 − I x.
I − A−1
C A−1 , x x∗
λ1
λn
1.4
If λ1 λn > 0, then,
Otherwise, λ1 λn < 0, then,
1
1
I − A−1
C A−1 , x x∗
A−1 − I x,
λk1
λk
1.5
λ1 ≤ · · · ≤ λk < 0 < λk1 ≤ · · · ≤ λn .
1.6
where
2. New Kantorovich Inequality for Hermite Matrices
Theorem 2.1. Let A be an n × n invertible Hermite matrix with real eigenvalues λ1 ≤ λ2 ≤ · · · ≤ λn .
Then
⎧
−1 ⎪
λ − λ1 2 ⎪
⎪ n
⎪
⎨ 4λ1 λn − CA, xC A , x
∗
x Axx∗ A−1 x − 1 ≤
⎪
⎪
λn − λ1 λk1 − λk ⎪
⎪
⎩
− CA, xC A−1 , x
4|λk λk1 |
if λ1 λn > 0,
2.1
if λ1 λn < 0
for any x ∈ Cn , x 1, where CA, x, CA−1 , x, λk , λk1 defined by 1.3, 1.4, 1.5, and 1.6.
Journal of Inequalities and Applications
3
To simplify the proof, we first introduce some lemmas.
Lemma 2.2. With the assumptions in Theorem 2.1, then
λ1 ≤ x∗ Ax ≤ λn ,
1
1
≤ x∗ A−1 x ≤ ,
λn
λ1
1
1
≤ x∗ A−1 x ≤
,
λk
λk1
0 ≤ λn − x∗ Axx∗ Ax − λ1 ≤
1
λn − λ1 2 ,
4
1
λn − λ1 2
∗ −1
0≤
x A x−
if λ1 λn > 0,
≤
λn
4λn λ1 2
1
1
λk1 − λk 2
∗ −1
∗ −1
0≤
−x A x
x A x−
if λ1 λn < 0
≤
λk1
λk
4λk1 λk 2
2.2
1
− x∗ A−1 x
λ1
for any x ∈ Cn , x 1.
Lemma 2.3. With the assumptions in Theorem 2.1, then
0 ≤ CA, x ≤
1
λn − λ1 2 ,
4
⎧
λ − λ1 2
⎪
⎪
⎪ n
⎪
⎨ 4λ1 λn 2
0 ≤ C A−1 , x ≤
⎪
⎪
λk1 − λk 2
⎪
⎪
⎩
4λk λk1 2
if λ1 λn > 0,
2.3
if λ1 λn < 0
for any x ∈ Cn , x 1.
Proof. Let x CA, x ki ϕi , then
ki ϕi λi − λ1 kj∗ ϕ∗j λn − λj
|ki |2 λn − λi λi − λ1 ≥ 0
2.4
while
λn λ1
λn − λ1
λn λ1
λn − λ1
I − A−
I
I A−
I
x
2
2
2
2
2
λn λ1
λn − λ1 2 λn − λ1 2
− A−
I x
.
≤
4
2
4
CA, x x∗
The proof about CA−1 , x is similar.
2.5
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Journal of Inequalities and Applications
Lemma 2.4. With the assumptions in Theorem 2.1, then
Ax2 − x∗ Ax2 ≤
λn − λ1 2
− CA, x,
4
⎧
λ − λ1 2
⎪
⎪
⎪ n
− C A−1 , x
⎪
2 2 ⎨ 4λn λ1 2
−1 A x − x∗ A−1 x ≤
⎪
⎪
λk1 − λk 2
⎪
⎪
− C A−1 , x
⎩
2
4λk1 λk if λ1 λn > 0,
2.6
if λ1 λn < 0
for any x ∈ Cn , x 1.
Proof. Thus,
Ax2 − x∗ Ax2 λn − x∗ Axx∗ Ax − λ1 x∗ A2 − λ1 λn A λ1 λn I x
λn − x∗ Axx∗ Ax − λ1 x∗ A − λ1 IA − λn Ix
λn − x∗ Axx∗ Ax − λ1 − x∗ A − λ1 Iλn I − Ax
≤
2.7
by Lemma 2.2 .
λn − λ1 2
− CA, x
4
The other inequality can be obtained similarly, the proof is completed.
We are now ready to prove the theorem.
Proof of the Theorem 2.1. Thus,
2 2
∗
x Axx∗ A−1 x − 1 x∗ x∗ A−1 x I − A−1 x∗ AxI − Ax
2
≤ x∗ A−1 x I − A−1 x x∗ AxI − Ax2 ,
2.8
while
x∗ AxI − Ax2 x∗ x∗ Ax2 I − 2x∗ AxA A2 x
x∗ A2 x − x∗ Ax2
2.9
Ax2 − x∗ Ax2 .
By the Lemma 2.4, we have
x∗ AxI − Ax2 ≤
λn − λ1 2
− CA, x.
4
2.10
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5
Similarly, we can get that,
⎧
λ − λ1 2
⎪
⎪
⎪ n
− C A−1 , x
⎪
2 2 2 ⎨ 4λn λ1 2
x∗ A−1 x I − A−1 x A−1 x − x∗ A−1 x ≤
⎪
⎪
λk1 − λk 2
⎪
⎪
− C A−1 , x
⎩
2
4λk1 λk if λ1 λn > 0,
if λ1 λn < 0.
2.11
Therefore,
⎧
−1 ⎪
λn − λ1 2
λn − λ1 2
⎪
⎪
− CA, x
− C A ,x
⎪
2
4
⎨
2 ⎪
4λ
λ
n
1
∗
x Axx∗ A−1 x − 1 ≤ ⎪
2
2
⎪
−
λ
−
λ
λ
λ
⎪
n
1
k1
k
−1
⎪
⎪
− CA, x
− C A ,x
⎩
4
4λk1 λk 2
if λ1 λn > 0,
if λ1 λn < 0.
2.12
From a2 − b2 c2 − d2 ≤ ac − bd2 for real numbers a, b, c, d, we have
⎧
2
⎪
−1 ⎪
λn − λ1 2 ⎪
⎪
− CA, xC A , x
,
if λ1 λn > 0
⎨
2 ⎪
4λ1 λn
∗
∗ −1
x Axx A x − 1 ≤
⎪
⎪
⎪
2
λn − λ1 λk1 − λk ⎪
⎪
, if λ1 λn < 0.
− CA, xC A−1 , x
⎩
4|λk λk1 |
2.13
The proof of Theorem 2.1 is completed.
Remark 2.5. Let A ∈ Mn be a positive definite Hermite matrix. From Theorem 2.1, we have
0 ≤ x∗ Axx∗ A−1 x − 1 ≤
λn − λ1 2
λn − λ1 2 − CA, xC A−1 , x ≤
,
4λ1 λn
4λ1 λn
2.14
our result improves the Kantorovich inequality 1.2, so we conclude that Theorem 2.1 gives
an improvement of the Kantorovich inequality that applies all invertible Hermite matrices.
Example 2.6. Let
⎞
1
0
0
⎟
⎜2
⎟
⎜
⎟
⎜
1⎟
1
⎜
⎜0
− ⎟,
⎜
2
4⎟
⎟
⎜
⎝
1 3 ⎠
0 −
4 8
⎛
⎛
⎞
2 0 0
⎜
⎟
A ⎝0 3 2⎠,
0 2 4
A−1
1
x √ 1, 1, 0T .
2
2.15
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Journal of Inequalities and Applications
The eigenvalues of A are: λ1 7 −
have
x∗ Axx∗ A−1 x − 1 1
,
4
√
√
17/2, λ2 2, λ3 7 17/2 by easily calculating, we
λ3 − λ1 2 17
,
4λ1 λ3
32
1
CA, xC A−1 , x .
32
2.16
Therefore,
λ3 − λ1 2
λ3 − λ1 2 − CA, xC A−1 , x <
,
4λ1 λ3
4λ1 λ3
2.17
we get a sharpen upper bound.
3. Conclusion
In this paper, we introduce a new Kantorovich type inequality for the invertible Hermite
matrices. In Theorem 2.1, if λ1 > 0, λn > 0, the result is well-known Kantorovich inequality.
Moreover, it holds for negative definite Hermite matrices, even for any invertible Hermite
matrix, there exists a similar inequality.
Acknowledgments
The authors would like to thank the two anonymous referees for their valuable comments
which have been implemented in this revised version. This work is supported by Natural
Science Foundation of Jiangxi, China No 2007GZS1760 and scienctific and technological
project of Jiangxi education office, China No GJJ08432.
References
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2 A. S. Householder, The Theory of Matrices in Numerical Analysis, Blaisdell, New York, NY, USA, 1964.
3 S.-G. Wang, “A matrix version of the Wielandt inequality and its applications to statistics,” Linear
Algebra and its Applications, vol. 296, no. 1–3, pp. 171–181, 1999.
4 J. Nocedal, “Theory of algorithms for unconstrained optimization,” Acta Numerica, vol. 1, pp. 199–242,
1992.