# Relative and Absolute Perturbation Bounds for Weighted Polar Decomposition

Authors: Pingping Zhang; Hu Yang; Hanyu Li
Journal: Journal of Applied Mathematics
 (2012)

---

## Abstract

Some new perturbation bounds for both weighted unitary polar factors and generalized nonnegative polar factors of the weighted polar decompositions are presented without the restriction thatA and its perturbed matrix A˜ have the same rank. These bounds improve the corresponding recent results.


---

## Body

## 1. Introduction

LetCm×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 R(A),r(A),A*,AMN#=N-1A*M,AMN†,∥A∥ and ∥A∥F 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 AMN# and AMN† can be found in details in [1, 2]. The weighted polar decomposition (MN-WPD) of A∈Cm×n is given by(1.1)A=QH,
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.Yang and Li [5] proved that the MN-WPD is unique under the condition(1.2)R(QMN#)=R(H).In this paper, we always assume that the MN-WPD satisfies condition (1.2).IfM=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 r(A)=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 for (generalized) polar decomposition without the restriction that A and Ã have the same rank.LetA∈Crm×n,Ã=A+E∈Csm×n have the (generalized) polar decompositions A=QH and Ã=Q̃H̃. For the perturbation bound of the (subunitary) unitary polar factors, the following two results can be found in [16](1.3)‖Q̃-Q‖F≤1min⁡{σr,σ̃s}‖E‖F,(1.4)‖Q̃-Q‖F≤22‖A†E‖F2+‖EA†‖F2+‖Ã†E‖F2+‖EÃ†‖F2.For the nonnegative polar factors, the perturbation bounds obtained by Chen [17] are(1.5)‖H̃-H‖≤(σ1+σ̃1σr+σ̃s+2)‖E‖,(1.6)‖H̃-H‖≤σ1σ̃1σ1+σ̃1(‖EA†‖+‖EÃ†‖)+σ1‖A†E‖+σ̃1‖Ã†E‖.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 Section2, 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.


## 2. Notation and Some Lemmas

Firstly, we introduce the definitions of the weighted norms.Definition 2.1.
LetA∈Cm×n. The norms ∥A∥(MN)=∥M1/2AN-1/2∥ and ∥A∥F(MN)=∥M1/2AN-1/2∥F are called the weighted unitarily invariant norm and weighted Frobenius norm of A, respectively. The definitions of ∥A∥(MN) and ∥A∥F(MN) can be also found in [20, 21].
LetA∈Crm×n and Ã∈Csm×n have their weighted singular value decompositions (MN-SVDs):
(2.1)A=UΣV*=(U1,U2)(Σ1000)(V1,V2)*=U1Σ1V1*,(2.2)Ã=ŨΣ̃Ṽ*=(Ũ1,Ũ2)(Σ̃1000)(Ṽ1,Ṽ2)*=Ũ1Σ̃1Ṽ1*.
Then the MN-WPDs of A=QH and Ã=Q̃H̃ can be obtained by
(2.3)Q=U1V1*,H=N-1V1Σ1V1*,Q̃=Ũ1Ṽ1*,H̃=N-1Ṽ1Σ̃1Ṽ1*,
where U=(U1,U2),Ũ=(Ũ1,Ũ2)∈Cm×m and V=(V1,V2),Ṽ=(Ṽ1,Ṽ2)∈Cn×n satisfy U*MU=Ũ*MŨ=Im and V*N-1V=Ṽ*N-1Ṽ=In, and U1∈Crm×r,Ũ1∈Csm×s,V1∈Crn×r,Ṽ1∈Csn×s,Σ1=diag(σ1,σ2,…,σr) and Σ̃1=diag(σ̃1,σ̃2,…,σ̃s). Here σ1≥σ2≥⋯≥σr>0 and σ̃1≥σ̃2≥⋯≥σ̃s>0 are the nonzero (M,N) weighted singular values of A and Ã, respectively.The following three lemmas can be found from [22], [23] and [16], respectively.Lemma 2.2.
LetB1 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 B1X-XB2=P and, moreover,
(2.4)‖X‖≤1η‖P‖.Lemma 2.3.
LetΩ∈Cs×s and Γ∈Ct×t be two Hermitian matrices, and let E,F∈Cs×t. If λ(Ω)∩λ(Γ)=⌀, then ΩX-XΓ=ΩE+FΓ has a unique solution X∈Cs×t, and, moreover,
(2.5)‖X‖F≤1η‖E‖F2+‖F‖F2,
where η=minλ∈λ(Ω),λ̃∈λ(Γ)(|λ-λ̃|/|λ̃|2+|λ|2).Lemma 2.4.
LetS=(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
(2.6)‖B‖F2=‖S1*BT1‖F2+‖S1*BT2‖F2+‖S2*BT2‖F2+‖S2*BT2‖F2.


## 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.Theorem 3.1.
LetA∈Crm×n and Ã=A+E∈Csm×n, and let A=QH and Ã=Q̃H̃ be their MN-WPDs of A and Ã, respectively. Then
(3.1)‖Q̃-Q‖F(MN)≤1min⁡{σr,σ̃s}‖E‖F(MN).Proof.
By (2.1), and (2.2) the perturbation E can be written as
(3.2)E=Ã-A=Ũ1Σ̃1Ṽ1*-U1Σ1V1*,
which together with the facts that U1*MU1=V1*N-1V1=Ir and Ũ1*MŨ1=Ṽ1*N-1Ṽ1=Is gives
(3.3)U1*MEN-1Ṽ1=U1*MŨ1Σ̃1-Σ1V1*N-1Ṽ1,(3.4)Ũ1*MEN-1V1=Σ̃1Ṽ1*N-1V1-Ũ1*MU1Σ1,(3.5)Ũ2*MEN-1V1=-Ũ2*MU1Σ1,U2*MEN-1Ṽ1=U2*MŨ1Σ̃1,(3.6)Ũ1*MEN-1V2=Σ̃1Ṽ1*N-1V2,U1*MEN-1Ṽ2=-Σ1V1*N-1Ṽ2.
Taking the conjugate transpose on both sides of (3.4) and subtracting it from (3.3) leads to
(3.7)Σ1(U1*MŨ1-V1*N-1Ṽ1)+(U1*MŨ1-V1*N-1Ṽ1)Σ̃1=U1*MEN-1Ṽ1-V1*N-1E*MŨ1.
Applying Lemma 2.2 to (3.7) for the Frobenius norm leads to
(3.8)‖U1*MŨ1-V1*N-1Ṽ1‖F≤1σr+σ̃s(‖U1*MEN-1Ṽ1‖F+‖V1*N-1E*MŨ1‖F).
Since
(3.9)U*M(Q̃-Q)N-1Ṽ=(U1*MŨ1-V1*N-1Ṽ1-V1*N-1Ṽ2U2*MŨ10),Ũ*M(Q̃-Q)N-1V=(Ṽ1*N-1V1-Ũ1*MU1Ṽ1*N-1V2-Ũ2*MU10),
it follows from Definition 2.1 and the fact that U*M1/2, Ũ*M1/2, N-1/2V, and N-1/2Ṽ are all unitary matrices that
(3.10)‖Q̃-Q‖F(MN)2=‖U*M(Q̃-Q)N-1Ṽ‖F2=‖U1*MŨ1-V1*N-1Ṽ1‖F2+‖V1*N-1Ṽ2‖F2+‖U2*MŨ1‖F2,(3.11)‖Q̃-Q‖F(MN)2=‖Ũ*M(Q̃-Q)N-1V‖F2=‖Ṽ1*N-1V1-Ũ1*MU1‖F2+‖Ṽ1*N-1V2‖F2+‖Ũ2*MU1‖F2.
Adding (3.10) to (3.11) gives
(3.12)2‖Q̃-Q‖F(MN)2=‖U1*MŨ1-V1*N-1Ṽ1‖F2+‖V1*N-1Ṽ2‖F2+‖U2*MŨ1‖F2+‖Ṽ1*N-1V1-Ũ1*MU1‖F2+‖Ṽ1*N-1V2‖F2+‖Ũ2*MU1‖F2=2‖U1*MŨ1-V1*N-1Ṽ1‖F2+‖V1*N-1Ṽ2‖F2+‖Ṽ1*N-1V2‖F2+‖U2*MŨ1‖F2+‖Ũ2*MU1‖F2.
Combing (3.5), (3.6), (3.8), (3.12), Lemma 2.4, and the fact that ∥U1*MŨ1-V1*N-1Ṽ1∥F2=∥Ṽ1*N-1V1-Ũ1*MU1∥F2 gets
(3.13)2‖Q̃-Q‖F(MN)2≤2(1σr+σ̃s)2(‖U1*MEN-1Ṽ1‖F+‖V1*N-1E*MŨ1‖F)2+1σr2‖U1*MEN-1Ṽ2‖F2+1σ̃s2‖U2*MEN-1Ṽ1‖F2+1σ̃s2‖Ũ1*MEN-1V2‖F2+1σr2‖Ũ2*MEN-1V1‖F2≤(2σr+σ̃s)2(‖U1*MEN-1Ṽ1‖F2+‖V1*N-1E*MŨ1‖F2)+1min⁡{σr2,σ̃s2}(‖U1*MEN-1Ṽ2‖F2+‖U2*MEN-1Ṽ1‖F2+‖Ũ1*MEN-1V2‖F2+‖Ũ2*MEN-1V1‖F2)≤1min⁡{σr2,σ̃s2}((‖U1*MEN-1Ṽ1‖F2+‖U1*MEN-1Ṽ2‖F2+‖U2*MEN-1Ṽ1‖F2)+(‖Ũ1*MEN-1V1‖F2+‖Ũ1*MEN-1V2‖F2+‖Ũ2*MEN-1V1‖F2))≤2min⁡{σr2,σ̃s2}‖M1/2EN-1/2‖F2=2min⁡{σr2,σ̃s2}‖E‖F(MN)2,
which proves the theorem.Remark 3.2.
IfM=Im and N=In in Theorem 3.1, the bound (3.1) is reduced to bound (1.3).Theorem 3.3.
LetA∈Crm×n and Ã=A+E∈Csm×n, and let A=QH and Ã=Q̃H̃ be their MN-WPDs of A and Ã, respectively. Then
(3.14)‖Q̃-Q‖F(MN)≤22‖ÃMN†E‖F(NN)2+‖AMN†E‖F(NN)2+‖EAMN†‖F(MM)2+‖EÃMN†‖F(MM)2.Proof.
From the MN-SVDs ofA and Ã in (2.1) and (2.2) and the facts that U*MU=Ũ*MŨ=Im and V*N-1V=Ṽ*N-1Ṽ=In, the weighted Moore-Penrose inverses of A and Ã can be written as
(3.15)AMN†=N-1V1Σ1-1U1*M,ÃMN†=N-1Ṽ1Σ̃1-1Ũ1*M.
Premultiplying the equation Ã-A=E by AMN† leads to
(3.16)AMN†Ã-AMN†A=AMN†E,
that is,
(3.17)N-1V1Σ1-1U1*MŨ1Σ̃1Ṽ1*-N-1V1V1*=AMN†E.
By (3.17), we can obtain
(3.18)U1*MŨ1Σ̃1-Σ1V1*N-1Ṽ1=Σ1V1*AMN†EN-1Ṽ1,-V1*N-1Ṽ2=V1*AMN†EN-1Ṽ2.
Similarly, by ÃMN†Ã-ÃMN†A=ÃMN†E, ÃAMN†-AAMN†=EAMN† and ÃÃMN†-AÃMN†=EÃMN†, we get
(3.19)Σ̃1Ṽ1*N-1V1-Ũ1*MU1Σ1=Σ̃1Ṽ1*ÃMN†EN-1V1,Ṽ1*N-1V2=Ṽ1*ÃMN†EN-1V2,(3.20)Σ̃1Ṽ1*N-1V1-Ũ1*MU1Σ1=Ũ1*MEAMN†U1Σ1,-Ũ2*MU1=Ũ2*MEAMN†U1,(3.21)U1*MŨ1Σ̃1-Σ1V1*N-1Ṽ1=U1*MEÃMN†Ũ1Σ̃1,U2*MŨ1=U2*MEÃMN†Ũ1,
respectively. By the first equations in (3.18)–(3.21), we derive
(3.22)(U1*MŨ1-V1*N-1Ṽ1)Σ̃1+Σ1(U1*MŨ1-V1*N-1Ṽ1)=Σ1V1*AMN†EN-1Ṽ1-(Ṽ1*ÃMN†EN-1V1)*Σ̃1,(3.23)(U1*MŨ1-V1*N-1Ṽ1)Σ̃1+Σ1(U1*MŨ1-V1*N-1Ṽ1)=U1*MEÃMN†Ũ1Σ̃1-Σ1(Ũ1*MEAMN†U1)*.
Applying Lemma 2.3 to (3.22) and (3.23), respectively, and noting that
(3.24)η=min⁡1≤i≤s,1≤j≤rσ̃i+σjσ̃i2+σj2≥1,
we find that
(3.25)‖U1*MŨ1-V1*N-1Ṽ1‖F2≤‖V1*AMN†EN-1Ṽ1‖F2+‖Ṽ1*ÃMN†EN-1V1‖F2,(3.26)‖U1*MŨ1-V1*N-1Ṽ1‖F2≤‖U1*MEÃMN†Ũ1‖F2+‖Ũ1*MEAMN†U1‖F2.
From (3.12), the second equations in (3.18)–(3.21), (3.25), (3.26), and Lemma 2.4, we deduce that
(3.27)2‖Q̃-Q‖F(MN)2≤‖V1*AMN†EN-1Ṽ1‖F2+‖Ṽ1*ÃMN†EN-1V1‖F2+‖U1*MEÃMN†Ũ1‖F2+‖Ũ1*MEAMN†U1‖F2+‖V1*AMN†EN-1Ṽ2‖F2+‖U2*MEÃMN†Ũ1‖F2+‖Ṽ1*ÃMN†EN-1V2‖F2+‖Ũ2*MEAMN†U1‖F2=(‖V1*AMN†EN-1Ṽ1‖F2+‖V1*AMN†EN-1Ṽ2‖F2)+(‖Ṽ1*ÃMN†EN-1V1‖F2+‖Ṽ1*ÃMN†EN-1V2‖F2)+(‖U1*MEÃMN†Ũ1‖F2+‖U2*MEÃMN†Ũ1‖F2)+(‖Ũ1*MEAMN†U1‖F2+‖Ũ2*MEAMN†U1‖F2)≤‖N1/2AMN†EN-1/2‖F2+‖N1/2ÃMN†EN-1/2‖F2+‖M1/2EÃMN†M-1/2‖F2+‖M1/2EAMN†M-1/2‖F2=‖AMN†E‖F(NN)2+‖ÃMN†E‖F(NN)2+‖EÃMN†‖F(MM)2+‖EAMN†‖F(MM)2,
which proves the theorem.Remark 3.4.
IfM=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.Theorem 4.1.
LetA∈Crm×n and Ã=A+E∈Csm×n, and let A=QH and Ã=Q̃H̃ be their MN-WPDs of A and Ã, respectively. Then
(4.1)‖H̃-H‖(NN)≤(σ1+σ̃1σr+σ̃s+2)‖E‖(MN).Proof.
By (2.1), (2.2), and (2.3), we have
(4.2)NH2=A*MA,NH̃2=Ã*MÃ,
which give
(4.3)NH̃(H̃-H)+N(H̃-H)H=Ã*M(Ã-A)+(Ã-A)*MA.
Let ΔH=H̃-H, we rewrite (4.3)
(4.4)NH̃ΔH+NΔHH=Ã*ME+E*MA,
that is,
(4.5)Ṽ1Σ̃1Ṽ1*ΔH+NΔHN-1V1Σ1V1*=Ṽ1Σ̃1Ũ1*ME+E*MU1Σ1V1*.
Premultiplying and postmultiplying both sides of (4.5) by, respectively, Ṽ1*N-1 and N-1V1 give
(4.6)Σ̃1Ṽ1*ΔHN-1V1+Ṽ1*ΔHN-1V1Σ1=Σ̃1Ũ1*MEN-1V1+Ṽ1*N-1E*MU1Σ1.
Similarly, we have
(4.7)Ṽ1*ΔHN-1V2=Ũ1*MEN-1V2,Ṽ2*ΔHN-1V1=Ṽ2*N-1E*MU1.
Applying Lemma 2.2 to (4.6) gives
(4.8)‖Ṽ1*ΔHN-1V1‖≤1σr+σ̃s‖Σ̃1Ũ1*MEN-1V1+Ṽ1*N-1E*MU1Σ1‖≤σ̃1σr+σ̃s‖Ũ1*MEN-1V1‖+σ1σr+σ̃s‖Ṽ1*N-1E*MU1‖≤σ̃1σr+σ̃s‖E‖(MN)+σ1σr+σ̃s‖E‖(MN)=σ̃1+σ1σr+σ̃s‖E‖(MN).
Notice that
(4.9)Ṽ*ΔHN-1V=(Ṽ1*ΔHN-1V1Ṽ1*ΔHN-1V2Ṽ2*ΔHN-1V10).
Combining (4.7)–(4.9) gives
(4.10)‖ΔH‖(NN)=‖Ṽ*ΔHN-1V‖≤‖Ṽ1*ΔHN-1V1‖+‖Ṽ1*ΔHN-1V2‖+‖Ṽ2*ΔHN-1V1‖≤σ̃1+σ1σr+σ̃s‖E‖(MN)+‖Ũ1*MEN-1V2‖+‖Ṽ2*N-1E*MU1‖≤σ̃1+σ1σr+σ̃s‖E‖(MN)+‖E‖(MN)+‖E‖(MN)=(σ̃1+σ1σr+σ̃s+2)‖E‖(MN),
which proves the theorem.Remark 4.2.
IfM=Im and N=In in Theorem 4.1, the bound (4.1) is reduced to bound (1.5).Ifr=n,s<n or s=n,r<n or r=s=n, we can easily derive the following three corollaries.Corollary 4.3.
LetA∈Cnm×n and Ã=A+E∈Csm×n, and let A=QH and Ã=Q̃H̃ be their MN-WPDs of A and Ã, respectively. Then
(4.11)‖H̃-H‖(NN)≤(σ1+σ̃1σn+σ̃s+1)‖E‖(MN).Corollary 4.4.
LetA∈Crm×n and Ã=A+E∈Cnm×n, and let A=QH and Ã=Q̃H̃ be their MN-WPDs of A and Ã, respectively. Then
(4.12)‖H̃-H‖(NN)≤(σ1+σ̃1σr+σ̃n+1)‖E‖(MN).Corollary 4.5.
LetA,Ã=A+E∈Cnm×n, and let A=QH and Ã=Q̃H̃ be their MN-WPDs of A and Ã, respectively. Then
(4.13)‖H̃-H‖(NN)≤σ1+σ̃1σn+σ̃n‖E‖(MN).If we take the weighted Frobenius norm as the specific weighted unitarily invariant norm in Theorem4.1, an alternative absolute perturbation bound can be derived as follows.Theorem 4.6.
LetA∈Crm×n and Ã=A+E∈Csm×n, and let A=QH and Ã=Q̃H̃ be their MN-WPDs of A and Ã, respectively. Then
(4.14)‖H̃-H‖F(NN)≤(2+2max⁡{σ12+σ̃s2σ1+σ̃s,σr2+σ̃12σr+σ̃1})‖E‖F(MN).Proof.
Applying Lemma2.3 to (4.6) gives
(4.15)‖Ṽ1*ΔHN-1V1‖F2≤max⁡1≤i≤r1≤j≤sσi2+σ̃j2(σi+σ̃j)2(‖Ũ1*MEN-1V1‖F2+‖Ṽ1*N-1E*MU1‖F2)≤2max⁡1≤i≤r1≤j≤sσi2+σ̃j2(σi+σ̃j)2‖E‖F(MN)2.
From [16], we know
(4.16)max⁡1≤i≤r1≤j≤s⁡σ̃i2+σ̃j2(σi+σ̃j)2=max⁡{σ12+σ̃s2(σ1+σ̃s)2,σr2+σ̃12(σr+σ̃1)2},
which together with (4.7), (4.9), and (4.15) gives
(4.17)‖ΔH‖F(NN)=‖Ṽ*ΔHN-1V‖F≤‖Ṽ1*ΔHN-1V1‖F+‖Ṽ1*ΔHN-1V2‖F+‖Ṽ2*ΔHN-1V1‖F≤2max⁡{σ12+σ̃s2σ1+σ̃s,σr2+σ̃12σr+σ̃1}‖E‖F(MN)+‖Ũ1*MEN-1V2‖F+‖Ṽ2*N-1E*MU1‖F≤2max⁡{σ12+σ̃s2σ1+σ̃s,σr2+σ̃12σr+σ̃1}‖E‖F(MN)+‖E‖F(MN)+‖E‖F(MN)=(2max⁡{σ12+σ̃s2σ1+σ̃s,σr2+σ̃12σr+σ̃1}+2)‖E‖F(MN).
Hence, we complete the theorem.Similarly, we can obtain the following three corollaries.Corollary 4.7.
LetA∈Cnm×n and Ã=A+E∈Csm×n, and let A=QH and Ã=Q̃H̃ be their MN-WPDs of A and Ã, respectively. Then
(4.18)‖H̃-H‖F(NN)≤(1+2max⁡{σ12+σ̃s2σ1+σ̃s,σn2+σ̃12σn+σ̃1})‖E‖F(MN).Corollary 4.8.
LetA∈Crm×n and Ã=A+E∈Cnm×n, and let A=QH and Ã=Q̃H̃ be their MN-WPDs of A and Ã, respectively. Then
(4.19)‖H̃-H‖F(NN)≤(1+2max⁡{σ12+σ̃n2σ1+σ̃n,σr2+σ̃12σr+σ̃1})‖E‖F(MN).Corollary 4.9.
LetA,Ã=A+E∈Cnm×n, and let A=QH and Ã=Q̃H̃ be their MN-WPDs of A and Ã, respectively. Then
(4.20)‖H̃-H‖F(NN)≤2max⁡{σ12+σ̃n2σ1+σ̃n,σn2+σ̃12σn+σ̃1}‖E‖F(MN).The relative perturbation bound for the generalized nonnegative polar factors is given in the following theorem.Theorem 4.10.
LetA∈Crm×n and Ã=A+E∈Csm×n, and let A=QH and Ã=Q̃H̃ be their MN-WPDs of A and Ã, respectively. Then
(4.21)‖H̃-H‖(NN)≤σ1σ̃1σ1+σ̃1(‖EAMN†‖(MM)+‖EÃMN†‖(MM))+σ1‖AMN†E‖(NN)+σ̃1‖ÃMN†E‖(NN).Proof.
From the proof of Theorem3.3, we know that
(4.22)AMN†=N-1V1Σ1-1U1*M,ÃMN†=N-1Ṽ1Σ̃1-1Ũ1*M.
Premultiplying and postmultiplying both sides of (4.5) by, respectively, (ÃMN†)* and AMN† give
(4.23)MŨ1Ṽ1*ΔHN-1V1Σ1-1U1*M+MŨ1Σ̃1-1Ṽ1*ΔHN-1V1U1*M=MŨ1Ũ1*MEN-1V1Σ1-1U1*M+MŨ1Σ̃1-1Ṽ1*N-1E*MU1U1*M=MŨ1Ũ1*MEAMN†+(EÃMN†)*MU1U1*M.
Premultiplying and postmultiplying both sides of (4.23) by, respectively, Ũ1* and U1 give
(4.24)Ṽ1*ΔHN-1V1Σ1-1+Σ̃1-1Ṽ1*ΔHN-1V1=Ũ1*MEAMN†U1+Ũ1*(EÃMN†)*MU1,
which together with Lemma 2.2 gives
(4.25)‖Ṽ1*ΔHN-1V1‖≤σ1σ̃1σ1+σ̃1(‖Ũ1*MEAMN†U1‖+‖Ũ1*(EÃMN†)*MU1‖)≤σ1σ̃1σ1+σ̃1(‖EAMN†‖(MM)+‖EÃMN†‖(MM)).
By (4.7) and the facts that AAMN†=U1U1*M and ÃÃMN†=Ũ1Ũ1*M, we have
(4.26)Ṽ1*ΔHN-1V2=Ũ1*MEN-1V2=Ũ1*MŨ1Ũ1*MEN-1V2=Ũ1*MÃÃMN†EN-1V2=Σ̃1Ṽ1*ÃMN†EN-1V2,Ṽ2*ΔHN-1V1=Ṽ2*N-1E*MU1=Ṽ2*N-1E*MU1U1*MU1=Ṽ2*N-1(AMN†E)*A*MU1=Ṽ2*N-1(AMN†E)*V1Σ1.
It follows from (4.9), (4.25) and (4.26) that
(4.27)‖ΔH‖(NN)=‖N1/2ΔHN-1/2‖=‖Ṽ*ΔHN-1V‖≤‖Ṽ1*ΔHN-1V1‖+‖Ṽ1*ΔHN-1V2‖+‖Ṽ2*ΔHN-1V1‖≤σ1σ̃1σ1+σ̃1(‖EAMN†‖(MM)+‖EÃMN†‖(MM))+‖Σ̃1Ṽ1*ÃMN†EN-1V2‖+‖Ṽ2*N-1(AMN†E)*V1Σ1‖≤σ1σ̃1σ1+σ̃1(‖EAMN†‖(MM)+‖EÃMN†‖(MM))+σ̃1‖ÃMN†E‖(NN)+σ1‖AMN†E‖(NN),
which proves the theorem.Remark 4.11.
IfM=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.Corollary 4.12.
LetA∈Cnm×n and Ã=A+E∈Csm×n, and let A=QH and Ã=Q̃H̃ be their MN-WPDs of A and Ã, respectively. Then
(4.28)‖H̃-H‖(NN)≤σ1σ̃1σ1+σ̃1(‖EAMN†‖(MM)+‖EÃMN†‖(MM))+σ1‖AMN†E‖(NN).Corollary 4.13.
LetA∈Crm×n and Ã=A+E∈Cnm×n, and let A=QH and Ã=Q̃H̃ be their MN-WPDs of A and Ã, respectively. Then
(4.29)‖H̃-H‖(NN)≤σ1σ̃1σ1+σ̃1(‖EAMN†‖(MM)+‖EÃMN†‖(MM))+σ̃1‖ÃMN†E‖(NN).Corollary 4.14.
LetA∈Cnm×n and Ã=A+E∈Cnm×n, and let A=QH and Ã=Q̃H̃ be their MN-WPDs of A and Ã, respectively. Then
(4.30)‖H̃-H‖(NN)≤σ1σ̃1σ1+σ̃1(‖EAMN†‖(MM)+‖EÃMN†‖(MM)).


## 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.



---

*Source: 219025-2012-02-29.xml*