Electronic Journal of Linear Algebra
Volume 17 ELA Volume 17 (2008)
Article 35
2008
A parameterized lower bound for the smallest
singular value
Wei Zhang
[email protected]
Zheng-Zhi Han
Shu-Qian Shen
Follow this and additional works at: http://repository.uwyo.edu/ela
Recommended Citation
Zhang, Wei; Han, Zheng-Zhi; and Shen, Shu-Qian. (2008), "A parameterized lower bound for the smallest singular value", Electronic
Journal of Linear Algebra, Volume 17.
DOI: https://doi.org/10.13001/1081-3810.1278
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Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 17, pp. 487-492, October 2008
ELA
http://math.technion.ac.il/iic/ela
A PARAMETERIZED LOWER BOUND FOR THE SMALLEST
SINGULAR VALUE∗
WEI ZHANG† , ZHENG-ZHI HAN† , AND SHU-QIAN SHEN‡
Abstract. This paper presents a parameterized lower bound for the smallest singular value of
a matrix based on a new Geršgorin-type inclusion region that has been established recently by these
authors. The comparison of the new lower bound with known ones is supplemented with a numerical
example.
Key words. Eigenvalue, Inclusion region, Smallest singular value, Lower bound.
AMS subject classifications. 15A12, 65F15.
1. Introduction. To estimate matrix singular values is an attractive topic in
matrix theory and numerical analysis, especially to give a lower bound for the smallest
one. Let A = (aij ) ∈ Cn×n and let A∗ be the conjugate transpose of A. Then the
singular values of A are the square roots of the eigenvalues of AA∗ . Throughout
the paper we use σn (A) to denote the smallest singular value of A. Denote N :=
{1, 2, . . . , n}. Define, for all k ∈ N ,
rk (A) :=
j∈N \{k}
|akj |,
ck (A) :=
|ajk |,
hk (A) :=
j∈N \{k}
1
(rk (A) + ck (A)) .
2
In the past three decades, several useful lower bounds for the smallest singular
value of a matrix have been presented in the literature; see Varah [7] and Qi [6]. By
using Geršgorin’s theorem (see Chapter 6 of [1]), Johnson [4] obtained a lower bound
for σn (A):
(1.1)
σn (A) ≥ min {|akk | − hk (A)} .
k∈N
Recently, Johnson and Szulc [5] provided several further lower bounds for the
∗ Received
by the editors August 14, 2008. Accepted for publication September 26, 2008. Handling
Editor: Roger A. Horn.
† School of Electronic, Information and Electrical Engineering, Shanghai Jiao Tong University,
Shanghai, 200240, China ([email protected]).
‡ School of Applied Mathematics, University of Electronic Science and Technology of China,
Chengdu, 610054, China.
487
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 17, pp. 487-492, October 2008
ELA
http://math.technion.ac.il/iic/ela
488
Wei Zhang, Zheng-Zhi Han, and Shu-Qian Shen
smallest singular value. Two of them are
1
1
2 2
2
4|akk | + [rk (A) − ck (A)]
− 2hk (A)
(1.2)
σn (A) ≥ min
2 k∈N
and
(1.3)
σn (A) ≥
12 1
2
min |aii | + |akk | − (|aii | − |akk |) + 4hi (A)hk (A)
.
2 i=k
In this paper, after introducing a new inclusion region for eigenvalues of a matrix
in Section 2, we present a parameterized lower bound for the smallest singular value
in Section 3. In Section 4, a numerical example is given to compare our results with
the known ones.
2. Inclusion regions for eigenvalues. Recently, a new Geršgorin-type inclusion region for eigenvalues of a matrix has been provided by Huang, Zhang, and Shen
in [3]. To introduce the result, we first present some notation. Let S be a nonempty
subset of N . Denote S̄ := N \S and let P(N ) denote the power set of N . For
A = (aij ) ∈ Cn×n , define, for all i ∈ N
riS (A) :=
|aik |,
riS̄ (A) :=
k∈S\{i}
cSi (A) :=
|aik |,
k∈S̄\{i}
|aki |,
k∈S\{i}
cS̄i (A) :=
|aki |.
k∈S̄\{i}
If S contains a single element, say S = {i0 }, then we let riS0 (A) = 0. Similarly
riS̄0 (A) = 0 if S̄ = {i0 }. We sometimes use riS (cSi , riS̄ , cS̄i ) to denote riS (A) (cSi (A),
riS̄ (A), cS̄i (A), respectively). Define, for all i ∈ S and j ∈ S̄,
GSi (A) := z ∈ C : |z − aii | ≤ riS , GS̄j (A) := z ∈ C : |z − ajj | ≤ rjS̄
and
GSi,j (A) := z ∈ C : z ∈
/ GSi (A) ∪ GS̄j (A), |z − aii | − riS |z − ajj | − rjS̄ ≤ riS̄ rjS .
Proposition 2.1. ([3]) Let A = (aij ) ∈ Cn×n . Then all the eigenvalues of A
are located in
 


GS (A) :=
GSi (A) ∪ 
GS̄j (A) ∪ 
GSi,j (A) .
i∈S
j∈S̄
i∈S,j∈S̄
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 17, pp. 487-492, October 2008
ELA
http://math.technion.ac.il/iic/ela
A Parameterized Lower Bound for the Smallest Singular Value
489
This result is interesting, but it can not be applied directly to estimate σn (A). So
we must deduce other inclusion regions. Let α ∈ [0, 1] be given. For i ∈ S, j ∈ S̄,
define the following regions in the complex plane:
α S
Ui,j
(A) := z ∈ C : |z − aii | − riS ≤ riS̄ rjS
,
S
(A) :=
Vi,j
1−α z ∈ C : |z − ajj | − rjS̄ ≤ riS̄ rjS
,
Proposition 2.2. Let A = (aij ) ∈ Cn×n . Then all the eigenvalues of A are
located in

 

S
S
Ui,j
(A) ∪ 
Vi,j
(A) .
K S (A) := 
i∈S,j∈S̄
i∈S,j∈S̄
S
Proof. It is sufficient to show that GS (A) ⊂ K S (A). Note that GSi (A) ⊆ Ui,j
(A)
S̄
S
S
and Gj (A) ⊆ Vi,j (A). Therefore, for any z ∈ G (A), if
z∈
GSi (A)
i∈S
or z ∈
GS̄j (A),
j∈S̄
then z ∈ K S (A). Otherwise, there exist i0 ∈ S and j0 ∈ S̄, such that
α 1−α
riS̄0 rjS0
|z − ai0 i0 | − riS0 |z − aj0 j0 | − rjS̄0 ≤ riS̄0 rjS0 = riS̄0 rjS0
which leads to
α
|z − ai0 i0 | − riS0 ≤ riS̄0 rjS0
1−α
or |z − aj0 j0 | − rjS̄0 ≤ riS̄0 rjS0
.
Hence
z ∈ UiS0 ,j0 (A) ∪ ViS0 ,j0 (A).
And then z ∈ K S (A).
3. Main results. In this section we use the inclusions derived in Section 2 to
estimate σn (A). Denote the Hermitian part of A by
H(A) := 12 (A + A∗ ).
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 17, pp. 487-492, October 2008
ELA
http://math.technion.ac.il/iic/ela
490
Wei Zhang, Zheng-Zhi Han, and Shu-Qian Shen
Let λmin (H(A)) be the smallest eigenvalue of H(A). It is known that λmin (H(A)) is a
lower bound for σn (A) [2, p.227]. Moreover, define for all i ∈ S, j ∈ S̄, and α ∈ [0, 1],
α
S
(A) : = |aii | − 12 riS (A) + cSi (A) − 14 riS̄ (A) + cS̄i (A) rjS (A) + cSj (A)
,
Pi,j
QSi,j (A) : = |ajj | −
1
2
1−α
rjS̄ (A) + cS̄j (A) − 14 riS̄ (A) + cS̄i (A) rjS (A) + cSj (A)
.
Theorem 3.1. Let A = (aij ) ∈ Cn×n . Then
S
(3.1)
σn (A) ≥ min Pi,j
(A), QSi,j (A) .
i∈S,j∈S̄
Proof. We first define a diagonal matrix D = diag(eiθ1 , . . . , eiθn ), where eiθk akk =
|akk | if akk = 0 and θk = 0 if akk = 0, k ∈ N . Since D is unitary, the singular values
of DA are the same as those of A. Consequently, we have
σn (A) = σn (DA) ≥ λmin (H(DA)).
(3.2)
Denote B = (bkl ) := H(DA) = 12 (DA + A∗ D∗ ). Thus, bkk = |akk |, for k ∈ N , and
bkl =
1 kθk
e akl + ālk e−kθk ,
2
for all k = l,
k, l ∈ N.
Since B is a Hermitian matrix, its eigenvalues are all real. Let λmin (B) denote
the smallest eigenvalue of B. Then, by using Proposition 2.2, λmin (B) must satisfy
at least one of the following conditions
α
λmin (B) ≥ |bii | − riS (B) − riS̄ (B)rjS (B) ,
i ∈ S, j ∈ S̄,
1−α
,
λmin (B) ≥ |bjj | − rjS̄ (B) − riS̄ (B)rjS (B)
It follows that
i ∈ S, j ∈ S̄.
α
|bii | − riS (B) − riS̄ (B)rjS (B) ,
i∈S,j∈S̄
1−α S̄
S̄
S
|bjj | − rj (B) − ri (B)rj (B)
.
λmin (B) ≥ min
By applying the triangle inequality, we have
α
|bii | − riS (B) − riS̄ (B)rjS (B)
α
= |aii | − riS (H(DA)) − riS̄ (H(DA)) rjS (H(DA))
α
≥ |aii | − 12 riS (A) + cSi (A) − 14 riS̄ (A) + cS̄i (A) rjS (A) + cSj (A)
S
= Pi,j
(A).
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 17, pp. 487-492, October 2008
ELA
http://math.technion.ac.il/iic/ela
491
A Parameterized Lower Bound for the Smallest Singular Value
Similarly, one can obtain
1−α
|bii | − riS̄ (B) − riS̄ (B)rjS (B)
≥ QSi,j (A).
Then from (3.2), we have
σn (A) ≥ min
i∈S,j∈S̄
S
Pi,j (A), QSi,j (A) .
Since the bound (3.1) holds for any nonempty S ∈ P(N ) and any α ∈ [0, 1], we
have obtain the following corollaries:
S
Corollary 3.2. σn (A) ≥ max max min Pi,j
(A), QSi,j (A) .
S∈P(N ) α∈[0,1] i∈S,j∈S̄
Corollary 3.3. ([4]) σn (A) ≥ min |aii | −
i∈N
1
2
(ri (A) + ci (A)) .
4. Numerical example. In this section, we give a numerical example to compare our bound (3.1) with known ones.
Example 4.1. Consider the following matrices

11

A1 = 4
3
5
12
4

6
−5 ,
13

18

A2 = 6
−6
2
15
−3

−5
8 ,
17

6

A3 = 2
2
2
9
−2

−1
1 .
−13
Table 1. Comparison of lower bounds for σn (A)
Matrix
σn (Ai )
S
α
(1.1)
(1.2)
(1.3)
(3.1)
A1
A2
A3
5.8446
9.6861
4.5433
{1}
{2}
{3}
0.57
0.56
0.10
2.0000
5.5000
2.5000
2.1803
6.1172
3.6921
2.4861
5.7827
2.3028
2.5886
5.7989
2.8377
From Table 1, we can see that bounds (1.2), (1.3), (3.1) are not comparable.
Note that the tightness of our bounds depend on the choice of S and α in which,
unfortunately, we do not find a method such that the derived bounds are optimal.
However, these parameters offer the possibility to optimize the estimation. We hope
that future research will propose a method to determine the parameters S and α that
can give tighter lower bounds.
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 17, pp. 487-492, October 2008
ELA
http://math.technion.ac.il/iic/ela
492
Wei Zhang, Zheng-Zhi Han, and Shu-Qian Shen
REFERENCES
[1] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, 1985.
[2] R. A. Horn and C. R. Johnson. Topics in Matrix Analysis. Cambridge University Press, Cambridge, 1991.
[3] T. Z. Huang, W. Zhang, and S. Q. Shen. Regions containing eigenvalues of a matrix. Electron.
J. Linear Algebra, 15:215–224, 2006.
[4] C. R. Johnson. A Gersgorin-type lower bound for the smallest singular value. Linear Algebra
Appl., 112:1–7, 1989.
[5] C. R. Johnson and T. Szulc. Further lower bounds of the smallest singular value. Linear Algebra
Appl., 272:69–179, 1998.
[6] L. Qi. Some simple estimates for singular values of a matrix. Linear Algebra Appl., 56:105–119,
1984.
[7] J. M. Varah. A lower bound for the smallest singular value. Linear Algebra Appl., 11:3–5, 1975.