Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 219025, 15 pages
doi:10.1155/2012/219025
Research Article
Relative and Absolute Perturbation Bounds for
Weighted Polar Decomposition
Pingping Zhang, Hu Yang, and Hanyu Li
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
Correspondence should be addressed to Pingping Zhang, zhpp04010248@126.com
Received 21 October 2011; Accepted 20 December 2011
Academic Editor: Juan Manuel Peña
Copyright q 2012 Pingping Zhang 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.
Some new perturbation bounds for both weighted unitary polar factors and generalized
nonnegative polar factors of the weighted polar decompositions are presented without the
have the same rank. These bounds improve the
restriction that A and its perturbed matrix A
corresponding recent results.
1. Introduction
Let Cm×n , Crm×n , C≥m , C>m , and In denote the set of m × n complex matrices, subset of Cm×n
consisting of matrices with rank r, set of the Hermitian nonnegative definite matrices
of order m, subset of C≥m consisting of positive-definite matrices and n × n unit matrix,
respectively. Without specification, we always assume that m > n >max{r, s} and the given
weight matrices M ∈ C>m , N ∈ C>n . For A ∈ Cm×n , we denote by RA, rA, A∗ , A#MN †
N −1 A∗ M, AMN , A and AF the column space, rank, conjugate transpose, weighted
conjugate transpose or adjoint, weighted Moore-Penrose inverse, unitarily invariant norm,
†
and Frobenius norm of A, respectively. The definitions of A#MN and AMN can be found in
details in 1, 2. The weighted polar decomposition MN-WPD of A ∈ Cm×n is given by
A QH,
1.1
where Q is an M, N weighted partial isometric matrix 3, 4 and H satisfies NH ∈ C≥n .
In this case, Q and H are called the M, N weighted unitary polar factor and generalized
nonnegative polar factor, respectively, of this decomposition.
2
Journal of Applied Mathematics
Yang and Li 5 proved that the MN-WPD is unique under the condition
#
RH.
R QMN
1.2
In this paper, we always assume that the MN-WPD satisfies condition 1.2.
If M Im and N In , then the MN-WPD is reduced to the generalized polar
decomposition and Q and H are reduced to the subunitary polar factor and nonnegative
polar factor, respectively. Further, if rA n, then the MN-WPD is just the polar
decomposition and Q and H are just the unitary polar factor and positive polar factor.
The problem on estimating the perturbation bounds for both polar decomposition and
generalized polar decomposition under the assumption that the matrix and its perturbed
matrix have the same rank 6–15 attracted most attention, and only some attention was
given without the restriction 16, 17. However, the arbitrary perturbation case seems
important in both theoretical and practical problems. Now we list some published bounds
have the same
for generalized polar decomposition without the restriction that A and A
rank.
A E ∈ Csm×n have the generalized polar decompositions A Let A ∈ Crm×n , A
Q
H.
For the perturbation bound of the subunitary unitary polar factors, the
QH and A
following two results can be found in 16
Q − Q ≤
F
1
EF ,
min{σr , σs }
√ 2
† 2 † 2
2
2
E EA .
A† EF EA† F A
Q − Q ≤
F
F
F
2
1.3
1.4
For the nonnegative polar factors, the perturbation bounds obtained by Chen 17 are
σ σ
1
1
2 E,
H − H ≤
σr σs
σ1 σ1 † †
† E.
H − H ≤
σ1 A† E σ1 A
EA EA
σ1 σ1
1.5
1.6
It is known that different elements of a vector are usually needed to be given some
different weights in practice e.g., the residual of the linear system, and the problems with
weights, such as weighted generalized inverses problem and weighted least square problem,
draw more and more attention, see, for example, 1, 2, 18, 19. As a generalization of the
generalized polar decomposition, MN-WPD may be useful for these problems. Therefore,
it is of interest to study MN-WPD and its related properties.
Our goal of this paper is mainly to generalize the perturbation bounds in 1.3–1.6 to
those for the weighted polar factors of the MN-WPDs in the corresponding weighted norms.
The rest of this paper is organized as follows.
In Section 2, we list notation and some lemmas which are useful in the sequel. In
Section 3, we present an absolute perturbation bound and a relative perturbation bound
for the weighted unitary polar factors, respectively, and some perturbation bounds for the
generalized nonnegative polar factors are also given in Section 4.
Journal of Applied Mathematics
3
2. Notation and Some Lemmas
Firstly, we introduce the definitions of the weighted norms.
Definition 2.1. Let A ∈ Cm×n . The norms AMN M1/2 AN −1/2 and AFMN M1/2 AN −1/2 F are called the weighted unitarily invariant norm and weighted Frobenius
norm of A, respectively. The definitions of AMN and AFMN can be also found in
20, 21.
∈ Csm×n have their weighted singular value decompositions MNLet A ∈ Crm×n and A
SVDs:
Σ1
0
Σ
1
∗
A UΣV U1 , U2
0
A UΣV ∗ U1 , U2 0
V1 , V2 ∗ U1 Σ1 V1∗ ,
0
0 ∗ ∗
V1 , V2 U1 Σ1 V1 .
0
2.1
2.2
Q
H
can be obtained by
Then the MN-WPDs of A QH and A
Q U1 V1∗ ,
H N −1 V1 Σ1 V1∗ ,
U
1 V ∗ ,
Q
1
N −1 V1 Σ
1 V ∗ ,
H
1
2.3
U
1, U
2 ∈ Cm×m and V V1 , V2 , V V1 , V2 ∈ Cn×n satisfy
where U U1 , U2 , U
∗ MU
Im and V ∗ N −1 V V ∗ N −1 V
1 ∈ Csm×s , V1 ∈
In , and U1 ∈ Crm×r , U
U∗ MU U
n×r n×s
σ1 , σ2 , . . . , σs . Here σ1 ≥ σ2 ≥ · · · ≥
Cr , V1 ∈ Cs , Σ1 diagσ1 , σ2 , . . . , σr and Σ1 diag
σr > 0 and σ1 ≥ σ2 ≥ · · · ≥ σs > 0 are the nonzero M, N weighted singular values of A and
respectively.
A,
The following three lemmas can be found from 22, 23 and 16, respectively.
Lemma 2.2. Let B1 and B2 be two Hermitian matrices and let P be a complex matrix. Suppose that
there are two disjoint intervals separated by a gap of width at least η, where one interval contains the
spectrum of B1 and the other contains that of B2 . If η > 0, then there exists a unique solution X to the
matrix equation B1 X − XB2 P and, moreover,
X ≤
1
P .
η
2.4
Lemma 2.3. Let Ω ∈ Cs×s and Γ ∈ Ct×t be two Hermitian matrices, and let E, F ∈ Cs×t . If λΩ ∩
λΓ ∅, then ΩX − XΓ ΩE FΓ has a unique solution X ∈ Cs×t , and, moreover,
XF ≤
1
η
E2F F2F ,
2 |λ|2 .
|λ − λ|/
|λ|
where η minλ∈λΩ, λ∈λΓ
2.5
4
Journal of Applied Mathematics
Lemma 2.4. Let S S1 , S2 ∈ Cm×m and T T1 , T2 ∈ Cn×n be both unitary matrices, where
S1 ∈ Cm×r , T1 ∈ Cn×s . Then for any matrix B ∈ Cm×n , one has
2 2 2 2
B2F S∗1 BT1 F S∗1 BT2 F S∗2 BT2 F S∗2 BT2 F .
2.6
3. Perturbation Bounds for the Weighted Unitary Polar Factors
In this section, we present an absolute perturbation bound and a relative perturbation bound
for the weighted unitary polar factors.
A E ∈ Csm×n , and let A QH and A
Q
H
be their
Theorem 3.1. Let A ∈ Crm×n and A
respectively. Then
MN-WPDs of A and A,
Q − Q
FMN
≤
1
EFMN .
min{σr , σs }
3.1
Proof. By 2.1, and 2.2 the perturbation E can be written as
−AU
1Σ
1 V ∗ − U1 Σ1 V ∗ ,
EA
1
1
3.2
∗ MU
1 V ∗ N −1 V1 Is
which together with the facts that U1∗ MU1 V1∗ N −1 V1 Ir and U
1
1
gives
1Σ
1 − Σ1 V ∗ N −1 V1 ,
U1∗ MEN −1 V1 U1∗ MU
1
3.3
∗ MEN −1 V1 Σ
∗ MU1 Σ1 ,
1 V ∗ N −1 V1 − U
U
1
1
1
3.4
∗ MEN −1 V1 −U
∗ MU1 Σ1 ,
U
2
2
∗ MEN −1 V2 Σ
1 V ∗ N −1 V2 ,
U
1
1
1Σ
1,
U2∗ MEN −1 V1 U2∗ MU
3.5
U1∗ MEN −1 V2 −Σ1 V1∗ N −1 V2 .
3.6
Taking the conjugate transpose on both sides of 3.4 and subtracting it from 3.3 leads to
1 − V ∗ N −1 V1 U∗ MU
1 − V ∗ N −1 V1 Σ
1.
1 U∗ MEN −1 V1 − V ∗ N −1 E∗ MU
Σ1 U1∗ MU
1
1
1
1
1
3.7
Applying Lemma 2.2 to 3.7 for the Frobenius norm leads to
∗ U1 MU1 − V1∗ N −1 V1 ≤
F
1 ∗
1
U1 MEN −1 V1 V1∗ N −1 E∗ MU
.
F
F
σr σs
3.8
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5
Since
∗
1 − V ∗ N −1 V1 −V ∗ N −1 V2
M
U
U
1
1
1
− Q N V U M Q
,
1
U2∗ MU
0
∗ MU1 V ∗ N −1 V2
∗ N −1 V1 − U
V
∗
−1
1
1
1
M Q
−Q N V ,
U
∗ MU1
−U
0
2
∗
−1
3.9
∗ M1/2 , N −1/2 V , and N −1/2 V are all
it follows from Definition 2.1 and the fact that U∗ M1/2 , U
unitary matrices that
2
Q − Q
2
− Q N −1 V U∗ M Q
F
2 ∗ −1 2 ∗ 2
1 − V ∗ N −1 V1 U1∗ MU
N
M
U
V
,
V
U
2
1
2
1
1
3.10
2
Q − Q
2
∗
− Q N −1 V U
M Q
F
2 2 2
∗ −1
∗
∗
MU1 V1 N V1 − U
V1∗ N −1 V2 U
2 MU1 .
1
3.11
FMN
F
FMN
F
F
F
F
F
Adding 3.10 to 3.11 gives
2
2Q
− Q
FMN
2 2 2
1 − V ∗ N −1 V1 1
U1∗ MU
V1∗ N −1 V2 U2∗ MU
1
F
F
F
2 2 2
∗ −1
∗ MU1 V ∗ N −1 V2 U
∗ MU1 V1 N V1 − U
2
1
1
F
F
F
2 2 2
∗ 2U1 MU1 − V1∗ N −1 V1 V1∗ N −1 V2 V1∗ N −1 V2 F
F
F
2
∗ 2 ∗
U2 MU1 U2 MU1 .
F
3.12
F
1 − V ∗ N −1 V1 2 Combing 3.5, 3.6, 3.8, 3.12, Lemma 2.4, and the fact that U1∗ MU
1
F
∗ MU1 2 gets
V ∗ N −1 V1 − U
1
1
2
2Q
− Q
FMN
F
≤2
2 2
1
∗
1
U1 MEN −1 V1 V1∗ N −1 E∗ MU
F
F
σr σs
1
1
2
2
2 U1∗ MEN −1 V2 2 U2∗ MEN −1 V1 F
F
σr
σs
2
2
1
1
∗
∗
−1
−1
U
2 U
MEN
V
MEN
V
2
1
2
1
F
F
σs
σr2
2 2 2
∗ −1 ∗ 2
∗
−1 ≤
U1 MEN V1 V1 N E MU1 F
F
σr σs
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Journal of Applied Mathematics
2 2
1
2 2 U1∗ MEN −1 V2 U2∗ MEN −1 V1 F
F
min σr , σs
2 2 ∗
∗
−1
−1
U
MEN
V
U
MEN
V
2
2
1
1
F
≤
F
2 2
∗
U1 MEN −1 V1 U1∗ MEN −1 V2 2
1
min σr2 , σs
F
F
2 2
∗
−1
U2∗ MEN −1 V1 U
MEN
V
1
1
F
F
2
∗
−1
U
1 MEN V2 F
2 ∗
−1
U
MEN
V
1
2
F
≤
2
1/2
−1/2 EN
M
2
2
2
min σr , σs
F
2
2 2 E2FMN ,
min σr , σs
3.13
which proves the theorem.
Remark 3.2. If M Im and N In in Theorem 3.1, the bound 3.1 is reduced to bound 1.3.
A E ∈ Csm×n , and let A QH and A
Q
H
be their
Theorem 3.3. Let A ∈ Crm×n and A
respectively. Then
MN-WPDs of A and A,
Q − Q
FMN
√ 2
2
2 †
†
† 2
† 2
≤
AMN E
EAMN EA
AMN E
MN FMM .
FNN
FNN
FMM
2
3.14
in 2.1 and 2.2 and the facts that U∗ MU U
∗ MU
Proof. From the MN-SVDs of A and A
can be
Im and V ∗ N −1 V V ∗ N −1 V In , the weighted Moore-Penrose inverses of A and A
written as
† N −1 V1 Σ
∗ M.
−1 U
A
1
1
MN
†
∗
AMN N −1 V1 Σ−1
1 U1 M,
3.15
− A E by A
Premultiplying the equation A
MN leads to
†
†
− A† A A† E,
AMN A
MN
MN
3.16
†
∗
−1
∗
∗
N −1 V1 Σ−1
1 U1 MU1 Σ1 V1 − N V1 V1 AMN E.
3.17
that is,
Journal of Applied Mathematics
7
By 3.17, we can obtain
1Σ
1 − Σ1 V ∗ N −1 V1 Σ1 V ∗ A† EN −1 V1 ,
U1∗ MU
1
1 MN
−V1∗ N −1 V2 V1∗ AMN EN −1 V2 .
†
3.18
†
† A
† E, AA
† − AA†
†
†
−A
† A A
Similarly, by A
MN
MN
MN
MN
MN EAMN and AAMN − AAMN † , we get
EA
MN
−1
∗ MU1 Σ1 Σ
†
1 V ∗ N −1 V1 − U
1 V ∗ A
Σ
1
1
1 MN EN V1 ,
∗ MU1 Σ1 U
∗ MEA† U1 Σ1 ,
1 V ∗ N −1 V1 − U
Σ
1
1
1
MN
1Σ
† U
1 − Σ1 V ∗ N −1 V1 U∗ MEA
U1∗ MU
1
1
MN 1 Σ1 ,
† EN −1 V2 ,
V1∗ N −1 V2 V1∗ A
MN
3.19
∗ MU1 U
∗ MEA† U1 ,
−U
2
2
MN
3.20
1 U∗ MEA
† U
U2∗ MU
2
MN 1 ,
3.21
respectively. By the first equations in 3.18–3.21, we derive
1 − V ∗ N −1 V1 Σ
1 − V ∗ N −1 V1
1 Σ1 U ∗ M U
U1∗ MU
1
1
1
∗
†
† EN −1 V1 Σ
1,
Σ1 V1∗ AMN EN −1 V1 − V1∗ A
MN
3.22
1 − V ∗ N −1 V1 Σ
1 − V ∗ N −1 V1
1 Σ1 U ∗ M U
U1∗ MU
1
1
1
∗
†
† U
∗
U1∗ MEA
MN 1 Σ1 − Σ1 U1 MEAMN U1 .
3.23
Applying Lemma 2.3 to 3.22 and 3.23, respectively, and noting that
η
min
1≤i≤s,1≤j≤r
σi σj
≥ 1,
σi2 σj2
3.24
we find that
2 2 2
∗ †
†
−1
EN
V
U1 MU1 − V1∗ N −1 V1 ≤ V1∗ AMN EN −1 V1 V1∗ A
1 ,
MN
3.25
2 2 2
∗
∗ †
† U
U1 MU1 − V1∗ N −1 V1 ≤ U1∗ MEA
MN 1 U1 MEAMN U1 .
3.26
F
F
F
F
F
F
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Journal of Applied Mathematics
From 3.12, the second equations in 3.18–3.21, 3.25, 3.26, and Lemma 2.4, we deduce
that
2
2Q
− Q
FMN
2 2
†
†
−1
≤ V1∗ AMN EN −1 V1 V1∗ A
EN
V
1
MN
F
F
2 2
∗
†
† U
U1∗ MEA
MN 1 F U1 MEAMN U1 F
2 2
†
† U
1
V1∗ AMN EN −1 V2 U2∗ MEA
MN
F
F
2 2
∗ †
∗
†
−1
V1 AMN EN V2 U2 MEAMN U1 F
F
2 2 †
†
V1∗ AMN EN −1 V1 V1∗ AMN EN −1 V2 F
F
2 2 ∗ †
∗ †
−1
−1
V1 AMN EN V1 V1 AMN EN V2 F
F
2 2 ∗
† U
† U1∗ MEA
MN 1 F U2 MEAMN U1 F
2 2 ∗
∗
†
†
U
MEA
U
U
MEA
U
1
1
2
1
MN
MN
F
F
2 2
1/2 †
1/2 †
−1/2 −1/2 ≤ N AMN EN
N AMN EN
F
F
2 2
1/2 †
1/2
†
−1/2 −1/2 M EAMN M
M EAMN M
F
F
2
2
2
†
†
† AMN E
AMN E
EAMN FNN
FNN
FMM
† 2
EAMN ,
3.27
FMM
which proves the theorem.
Remark 3.4. If M Im and N In in Theorem 3.3, the bound 3.14 is reduced to bound 1.4.
4. Perturbation Bounds for the Generalized Nonnegative Polar Factors
In this section, two absolute perturbation bounds and a relative perturbation bound for the
generalized nonnegative polar factors are given.
A E ∈ Csm×n , and let A QH and A
Q
H
be their
Theorem 4.1. Let A ∈ Crm×n and A
MN-WPDs of A and A, respectively. Then
H − H NN
≤
σ1 σ1
2 EMN .
σr σs
4.1
Proof. By 2.1, 2.2, and 2.3, we have
NH 2 A∗ MA,
2 A
∗ MA,
NH
4.2
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9
which give
∗
H
−H N H
−A A
−H H A
∗M A
− A MA.
NH
4.3
− H, we rewrite 4.3
Let ΔH H
∗ ME E∗ MA,
N HΔH
NΔHH A
4.4
∗ ME E∗ MU1 Σ1 V ∗ .
1 V ∗ ΔH NΔHN −1 V1 Σ1 V ∗ V1 Σ
1U
V1 Σ
1
1
1
1
4.5
that is,
Premultiplying and postmultiplying both sides of 4.5 by, respectively, V1∗ N −1 and N −1 V1
give
∗ MEN −1 V1 V ∗ N −1 E∗ MU1 Σ1 .
1U
1 V ∗ ΔHN −1 V1 V ∗ ΔHN −1 V1 Σ1 Σ
Σ
1
1
1
1
4.6
Similarly, we have
∗ MEN −1 V2 ,
V1∗ ΔHN −1 V2 U
1
V2∗ ΔHN −1 V1 V2∗ N −1 E∗ MU1 .
4.7
Applying Lemma 2.2 to 4.6 gives
∗
V1 ΔHN −1 V1 ≤
1 ∗
Σ1 U1 MEN −1 V1 V1∗ N −1 E∗ MU1 Σ1 σr σs
σ1 σ1 ∗
∗ −1 ∗
≤
U1 MEN −1 V1 V1 N E MU1 σr σs
σr σs
σ1
σ1
≤
EMN EMN
σr σs
σr σs
σ1 σ1
EMN .
σr σs
4.8
V1∗ ΔHN −1 V1 V1∗ ΔHN −1 V2
.
0
V ∗ ΔHN −1 V1
4.9
Notice that
V ∗ ΔHN −1 V 2
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Journal of Applied Mathematics
Combining 4.7–4.9 gives
ΔHNN V ∗ ΔHN −1 V ≤ V1∗ ΔHN −1 V1 V1∗ ΔHN −1 V2 V2∗ ΔHN −1 V1 σ1 σ1
∗ −1 ∗
∗
−1
≤
EMN U
1 MEN V2 V2 N E MU1 σr σs
σ1 σ1
≤
EMN EMN EMN
σ σs
r
σ1 σ1
2 EMN ,
σr σs
4.10
which proves the theorem.
Remark 4.2. If M Im and N In in Theorem 4.1, the bound 4.1 is reduced to bound 1.5.
If r n, s < n or s n, r < n or r s n, we can easily derive the following three
corollaries.
A E ∈ Csm×n , and let A QH and A
Q
H
be their
Corollary 4.3. Let A ∈ Cnm×n and A
respectively. Then
MN-WPDs of A and A,
H − H NN
≤
σ1 σ1
1 EMN .
σn σs
4.11
A E ∈ Cnm×n , and let A QH and A
Q
H
be their
Corollary 4.4. Let A ∈ Crm×n and A
MN-WPDs of A and A, respectively. Then
H − H NN
≤
σ1 σ1
1 EMN .
σr σn
4.12
Q
H
be their MN-WPDs of A
A E ∈ Cnm×n , and let A QH and A
Corollary 4.5. Let A, A
and A, respectively. Then
H − H NN
≤
σ1 σ1
EMN .
σn σn
4.13
If we take the weighted Frobenius norm as the specific weighted unitarily invariant
norm in Theorem 4.1, an alternative absolute perturbation bound can be derived as follows.
A E ∈ Csm×n , and let A QH and A
Q
H
be their
Theorem 4.6. Let A ∈ Crm×n and A
respectively. Then
MN-WPDs of A and A,
H − H ⎛
FNN
⎜
≤ ⎝2 √
⎫⎞
⎧
⎪
⎬
⎨ σ12 σs2
σr2 σ12 ⎪
⎟
2 max
,
⎠EFMN .
⎪
⎪
σ
σ
σ
σ
s
r
1 ⎭
⎩ 1
4.14
Journal of Applied Mathematics
11
Proof. Applying Lemma 2.3 to 4.6 gives
2
2 2 σi2 σj2 ∗ −1 ∗
∗
∗
−1
−1
V1 ΔHN V1 ≤ max 2 U1 MEN V1 F V1 N E MU1 F
F
1≤i≤r σ σ
j
i
1≤j≤s
σi2 σj2
2
≤ 2max 2 EFMN .
1≤i≤r σ σ
j
i
1≤j≤s
4.15
From 16, we know
2
σi2 σj2
σ1 σs2
σr2 σ12
max ,
max
,
2
1≤i≤r σ σ
σ1 σs 2 σr σ1 2
j
i
1≤j≤s
4.16
which together with 4.7, 4.9, and 4.15 gives
ΔHFNN V ∗ ΔHN −1 V F
≤ V1∗ ΔHN −1 V1 V1∗ ΔHN −1 V2 V2∗ ΔHN −1 V1 F
F
⎧
⎫
⎪
2
2
2
2⎪
⎨
⎬
σ
σ
σ
σ
√
s
r
1
1
≤ 2 max
,
EFMN
⎪
σr σ1 ⎪
⎩ σ1 σs
⎭
F
∗
∗ −1 ∗
−1
U
MEN
V
V
N
E
MU
2
1
2
1
F
F
4.17
⎧
⎫
⎪
2
2
2
2⎪
⎨
⎬
σ
σ
σ
σ
√
s
r
1
1
≤ 2 max
,
EFMN EFMN EFMN
⎪
σr σ1 ⎪
⎩ σ1 σs
⎭
⎧
⎫
⎞
⎛
⎪
2
2
2
2⎪
⎨
⎬
σ
σ
σ
σ
√
s
r
1
1
⎟
⎜
2⎠EFMN .
⎝ 2 max
,
⎪
σr σ1 ⎪
⎩ σ1 σs
⎭
Hence, we complete the theorem.
Similarly, we can obtain the following three corollaries.
A E ∈ Csm×n , and let A QH and A
Q
H
be their
Corollary 4.7. Let A ∈ Cnm×n and A
MN-WPDs of A and A, respectively. Then
H − H ⎛
FNN
⎜
≤ ⎝1 √
⎫⎞
⎧
⎪
⎬
⎨ σ12 σs2
σn2 σ12 ⎪
⎟
2 max
,
⎠EFMN .
⎪
⎪
σ
σ
σ
σ
s
n
1 ⎭
⎩ 1
4.18
12
Journal of Applied Mathematics
A E ∈ Cnm×n , and let A QH and A
Q
H
be their
Corollary 4.8. Let A ∈ Crm×n and A
respectively. Then
MN-WPDs of A and A,
H − H ⎛
FNN
⎜
≤ ⎝1 √
⎧
⎫⎞
⎪
⎨ σ12 σn2
⎬
σr2 σ12 ⎪
⎟
2 max
,
⎠EFMN .
⎪
⎪
σ
σ
σ
σ
1
n
r
1
⎩
⎭
4.19
A E ∈ Cnm×n , and let A QH and A
Q
H
be their MN-WPDs of A
Corollary 4.9. Let A, A
and A, respectively. Then
H − H FNN
≤
√
⎧
⎫
⎪
⎨ σ12 σn2
⎬
σn2 σ12 ⎪
2 max
,
EFMN .
⎪
σn σ1 ⎪
⎩ σ1 σn
⎭
4.20
The relative perturbation bound for the generalized nonnegative polar factors is given
in the following theorem.
A E ∈ Csm×n , and let A QH and A
Q
H
be their
Theorem 4.10. Let A ∈ Crm×n and A
respectively. Then
MN-WPDs of A and A,
H − H NN
≤
σ1 σ1 † † EA
EAMN MN MM
MM
σ1 σ1
†
†
σ1 AMN E
σ1 A
.
MN E
NN
4.21
NN
Proof. From the proof of Theorem 3.3, we know that
† N −1 V1 Σ
∗ M.
−1 U
A
1
1
MN
†
∗
AMN N −1 V1 Σ−1
1 U1 M,
4.22
∗
Premultiplying and postmultiplying both sides of 4.5 by, respectively, A
MN and AMN
give
†
†
1Σ
1 V ∗ ΔHN −1 V1 Σ−1 U∗ M MU
−1 V ∗ ΔHN −1 V1 U∗ M
MU
1
1
1
1
1
1
1U
1Σ
∗ MEN −1 V1 Σ−1 U∗ M MU
−1 V ∗ N −1 E∗ MU1 U∗ M
MU
1
1
1
1
1
1
∗
†
†
∗
∗
MEA
1U
EA
MU1 U M.
MU
1
MN
MN
4.23
1
∗ and U1 give
Premultiplying and postmultiplying both sides of 4.23 by, respectively, U
1
∗
†
−1
†
∗
∗
−1 ∗
MU1 ,
V1∗ ΔHN −1 V1 Σ−1
1 Σ1 V1 ΔHN V1 U1 MEAMN U1 U1 EAMN
4.24
Journal of Applied Mathematics
13
which together with Lemma 2.2 gives
∗
σ1 σ1 ∗
∗
∗
†
†
E
A
MU
V1 ΔHN −1 V1 ≤
U1 MEAMN U1 U
1
1
MN
σ1 σ1
σ1 σ1 † † ≤
EA
.
EAMN MN
MM
MM
σ1 σ1
4.25
A
∗
By 4.7 and the facts that AAMN U1 U1∗ M and A
MN U1 U1 M, we have
†
†
∗ MEN −1 V2 U
∗ MU
1U
∗ MEN −1 V2
V1∗ ΔHN −1 V2 U
1
1
1
−1
∗ MA
A
† EN −1 V2 Σ
†
1 V ∗ A
U
1
1 MN EN V2 ,
MN
V2∗ ΔHN −1 V1 V2∗ N −1 E∗ MU1 V2∗ N −1 E∗ MU1 U1∗ MU1
∗
∗
†
†
V2∗ N −1 AMN E A∗ MU1 V2∗ N −1 AMN E V1 Σ1 .
4.26
It follows from 4.9, 4.25 and 4.26 that
ΔHNN N 1/2 ΔHN −1/2 V ∗ ΔHN −1 V ≤ V1∗ ΔHN −1 V1 V1∗ ΔHN −1 V2 V2∗ ΔHN −1 V1 σ1 σ1 † † ≤
EA
EAMN MN MM
MM
σ1 σ1
∗
∗ −1 †
∗ †
−1
AMN E V1 Σ1 Σ
1 V1 AMN EN V2 V2 N
σ1 σ1 † † ≤
EA
EAMN MN MM
MM
σ1 σ1
†
†
σ1 A
E
σ
E
,
A
1
MN
MN NN
4.27
NN
which proves the theorem.
Remark 4.11. If M Im and N In in Theorem 4.10, the bound 4.21 is reduced to bound
1.6.
The following three corollaries can be also easily obtained.
A E ∈ Csm×n , and let A QH and A
Q
H
be their
Corollary 4.12. Let A ∈ Cnm×n and A
MN-WPDs of A and A, respectively. Then
H − H NN
σ1 σ1 †
† † ≤
EAMN .
σ1 AMN E
EAMN MM
MM
NN
σ1 σ1
4.28
A E ∈ Cnm×n , and let A QH and A
Q
H
be their
Corollary 4.13. Let A ∈ Crm×n and A
MN-WPDs of A and A, respectively. Then
H − H NN
σ1 σ1 †
† † ≤
EAMN E
σ1 A
EAMN MN NN .
MM
MM
σ1 σ1
4.29
14
Journal of Applied Mathematics
A E ∈ Cnm×n , and let A QH and A
Q
H
be their
Corollary 4.14. Let A ∈ Cnm×n and A
respectively. Then
MN-WPDs of A and A,
H − H NN
σ1 σ1 † † ≤
EAMN .
EAMN MM
MM
σ1 σ1
4.30
5. Conclusion
In this paper, we obtain the relative and absolute perturbation bounds for the weighted
polar decomposition without the restriction that the original matrix and its perturbed matrix
have the same rank. These bounds are the corresponding generalizations of those for the
generalized polar decomposition.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China no.
11171361 and in part by the Natural Science Foundation Project of CQ CSTC 2010BB9215.
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