THE OPTIMAL PERTURBATION BOUNDS FOR THE WEIGHTED MOORE-PENROSE INVERSE∗
WEI-WEI XU†, LI-XIA CAI‡, AND WEN LI§
Abstract. In this paper, we obtain optimal perturbation bounds of the weighted Moore-Penrose inverse under the weighted unitary invariant norm, the weightedQ-norm and the weightedF-norm, and thereby extend some recent results.
Key words. Weighted Moore-Penrose inverse, Weighted unitary invariant norm, Weighted Q-norm, WeightedF-norm.
AMS subject classifications. 15A09, 15A18, 15A24.
1. Introduction. LetCm×nbe the set of complexm×nmatrices andCm×nr be the subset consisting of all matrices in Cm×n of rankr. Let A∈Cm×n. We denote kAk, kAk2, kAkQ and kAkF by the unitary invariant norm, spectral norm, Q-norm andF-norm ofA, respectively. The conjugate transformation and the Moore-Penrose generalized inverse of a matrixAare denoted byA∗ andA†, respectively.
Weighted problems, such as the weighted generalized inverse problem and the weighted least squares problem, draw more and more attention, see e.g., [2, 4, 8, 12].
A generalization of the generalized inverse is the weighted Moore-Penrose inverse of an arbitrary matrix which has many applications in numerical computation, statistics, prediction theory, control systems and analysis and curve fitting, see e.g., [1, 9, 14].
There have been many numerical methods for the computation of the weighted Moore- Penrose inverse, see e.g., [6, 7, 10, 11]. It is an interesting problem to determine how the weighted Moore-Penrose inverse is transformed under perturbation. Answers to this problem will have application in numerical computation, prediction theory and curve fitting. Therefore, it is of significance to estimate the optimal perturbation bounds of the weighted Moore-Penrose inverse. The weighted unitary invariant norm
∗Received by the editors on October 17, 2010. Accepted for publication on May 14, 2011. Han- dling Editor: Miroslav Fiedler.
†Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, PO Box 2719, Beijing 100190, P.R. China ([email protected]).
‡School of Mathematical Sciences, South China Normal University, Guangzhou, 510631 China ([email protected]).
§School of Mathematical Sciences, South China Normal University, Guangzhou, 510631 China ([email protected]). The work was supported in part by Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20104407110001).
521
is a more general norm and in terms of this norm, the bounds for the weighted Moore- Penrose inverse can be characterized by weighted singular values ((M, N) singular values). Recently, much effort has been made for estimating perturbation bounds of the Moore-Penrose inverse, see e.g., [5, 9, 13, 14]. In [13], Wedin presented the perturbation bounds of the Moore-Penrose inverse under a general unitarily invariant norm, the spectral norm and Frobenius norm, respectively. Meng and Zheng in [5]
obtained the optimal perturbation bounds for the Moore-Penrose inverse under the Frobenius norm. Cai et al. in [3] obtained the additive and multiplicative perturbation bounds for the Moore-Penrose inverse under the unitary invariant norm and theQ- norm, which improves the corresponding results in [13]. In this paper, we will focus our attention on optimal perturbation bounds for the weighted Moore-Penrose inverse in the weighted unitary invariant norm, the weighted Q-norm and the weighted F- norm and thereby extend the corresponding results in [3] and [5].
We first introduce some basic definitions:
Definition 1.1. [3] A unitary invariant norm k · kis called a Q-norm if there exists another unitarily invariant norm k · k′ such that kYk = (kY∗Yk′)12, which is denoted byk · kQ.
Note thatF-norm and 2-norm areQ-norms.
Definition 1.2. [15] For an arbitrary matrix A ∈ Cm×n, there is a unique matrixX∈Cn×msatisfying the following equalities:
• AXA=A;
• XAX=X;
• (M AX)∗ =M AX;
• (N XA)∗=N XA.
Then matrix X is called a weighted Moore-Penrose inverse of A and denoted by X =A†M N. HereM andN are the given Hermitian positive definite matrices, which are called weighted matrices.
Definition 1.3. [15] LetA∈Cm×n. Then the following norms
• kAk(M N)=kM12AN−12k;
• kAkF(M N)=kM12AN−12kF;
• kAkQ(M N)=kM12AN−12kQ;
• kAkM N =kM12AN−12k2,
are called the weighted unitary invariant norm, the weightedF-norm, the weighted Q-norm and the weighted spectral norm ofA, respectively.
Definition 1.4. [15] Let A ∈ Cm×nr . The (M, N) weighted singular value de-
composition (MN-SVD) ofA∈Cm×nr is expressed as follows:
A=U
Σ 0 0 0
V∗=U1ΣV1∗, (1.1)
where U = (U1, U2) ∈ Cm×n and V = (V1, V2) ∈ Cn×n satisfy U∗M U = Im and V∗N−1V = In, Σ =diag(σ1, . . . , σr), σi = √
λi and λ1 ≥ · · · ≥ λr > 0 are the nonzero eigenvalues ofN−1A∗M A. Thenσ1, . . . , σr>0 are called the nonzero (M, N) weighted singular values ofA.
The rest of this paper is organized as follows. In Section 2, we give some lemmas, which are useful to deduce our main results. In Sections 3 and 4, we consider the additive and multiplicative perturbation of the weighted Moore-Penrose inverse. Some new bounds for additive and multiplicative perturbation under the normsk · k(M N), k · kQ(M N)andk · kF(M N)are presented, which extends the corresponding ones in [5]
and [13]. In Section 5, we give some numerical examples to illustrate the optimality of our given bounds under the weightedQ-norm andF-norm, respectively. Finally, in Section 6 we give concluding remarks.
2. Preliminaries. In this section we give some lemmas, which are useful to deduce our main results.
Lemma 2.1. [12] Let Ahave MN-SVD (1.1). Then (1)A†M N =N−1V1Σ−1U1∗M;
(2)kA†M NkN M = σ1
r.
Lemma 2.2. [3]Let B have the block form B=
B11 B12 B21 B22
.
Then
kBk2Q≤ kB11k2Q+kB12k2Q+kB21k2Q+kB22k2Q.
Lemma 2.3. [3]LetB1andB2be two Hermitian matrices and letP 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,
kXk ≤ 1 ηkPk.
Lemma 2.4. [3]Let W ∈Cn×n be a unitary matrix with the block form W =
W11 W12
W21 W22
,
where W11 ∈Cr×r,W22 ∈ C(n−r)×(n−r), 1 ≤r < n. ThenkW12k=kW21k for any unitarily invariant norm.
3. Additive perturbation bounds. In this section, we will present optimal ad- ditive perturbation bounds of the weighted Moore-Penrose inverse under the weighted unitarily invariant norm, the weightedQ-norm and the weightedF-norm, respectively.
Theorem 3.1. Let A∈Cm×nr andB=A+E∈Cm×ns . Then
kBM N† −A†M Nk(N M)≤(kA†M NkN MkBM N† kN M+ max{kA†M Nk2N M, kB†M Nk2N M})kEk(M N).
(3.1)
Proof. LetAandB have the following (M, N) weighted singular value decompo- sitions:
A=U
Σ1 0 0 0
V∗=U1Σ1V1∗, B= ˜U
Σ˜1 0 0 0
V˜∗= ˜U1Σ˜1V˜1∗, (3.2)
where U = (U1, U2), ˜U = ( ˜U1,U˜2) ∈ Cm×m, V = (V1, V2), ˜V = ( ˜V1, ˜V2) ∈ Cn×n satisfy U∗M U = Im, U˜∗MU˜ = Im, V∗N−1V = In and ˜V∗N−1V˜ = In, Σ1=diag(σ1, . . . , σr), Σ˜1 =diag(˜σ1, . . . ,σ˜s) with σ1 ≥ · · · ≥σr>0 and ˜σ1≥ · · · ≥
˜ σs>0.
By (3.2) we have
E=B−A= ˜U1Σ˜1V˜1∗−U1Σ1V1∗. (3.3)
By the MN-SVDs (3.2) ofA andB we know that M12U, M12U , N˜ −12V andN−12V˜ are unitary matrices. Hence from (3.3) one may deduce that
Σ˜1V˜1∗N−1V1−U˜1∗M U1Σ1= ˜U1∗M EN−1V1, (3.4)
U2∗MU˜1Σ˜1=U2∗M EN−1V˜1, (3.5)
Σ1V1∗N−1V˜2=−U1∗M EN−1V˜2. (3.6)
It follows from (3.4) that
V˜1∗N−1V1Σ−11 −Σ˜−11 U˜1∗M U1= ˜Σ−11 U˜1∗M EN−1V1Σ−11 . (3.7)
By Lemma 2.1 we obtainA†M N =N−1V1Σ−11 U1∗M andBM N† =N−1V˜1Σ˜−11 U˜1∗M.
Then, by Definition 1.3, we have
kBM N† −A†M Nk(N M)=kN12(N−1V˜1Σ˜−11 U˜1∗M −N−1V1Σ−11 U1∗M)M−12k
=kV˜∗N−12(N−12V˜1Σ˜−11 U˜1∗M12 −N−12V1Σ−11 U1∗M12)M12Uk
=kV˜∗(N−1V˜1Σ˜−11 U˜1∗M−N−1V1Σ−11 U1∗M)Uk
=
V˜1∗ V˜2∗
(N−1V˜1Σ˜−11 U˜1∗M −N−1V1Σ−11 U1∗M)(U1, U2)
=
Σ˜−11 U˜1∗M U1−V˜1∗N−1V1Σ−11 Σ˜−11 U˜1∗M U2
−V˜2∗N−1V1Σ−11 0
, (3.8)
from which one may deduce that kBM N† −A†M Nk(N M)≤
Σ˜−11 U˜1∗M U1−V˜1∗N−1V1Σ−11 0
0 0
+
0 Σ˜−11 U˜1∗M U2
−V˜2∗N−1V1Σ−11 0
. (3.9)
By (3.7) we have
Σ˜−11 U˜1∗M U1−V˜1∗N−1V1Σ−11 0
0 0
≤ 1
σrσ˜skM12EN−12k. (3.10)
By (3.5) and (3.6) we have
Σ˜−11 U˜1∗M U2= ˜Σ−21 (U2∗M EN−1V˜1)∗, V˜2∗N−1V1Σ−11 = (U1∗M EN−1V˜2)∗Σ−21 . Thus
0 Σ˜−11 U˜1∗M U2
−V˜2∗N−1V1Σ−11 0
(3.11)
≤max{ 1 σ2r, 1
˜ σ2s}
0 (U2∗M EN−1V˜1)∗
−(U1∗M EN−1V˜2)∗ 0
. Notice that
0 (U2∗M EN−1V˜1)∗ (U1∗M EN−1V˜2)∗ 0
≤
V˜1∗ V˜2∗
N−1E∗M(U1, U2)
=V˜∗N−1E∗M U . (3.12)
SinceM12U andN−12V˜ are unitary matrices, it follows from (3.11) and (3.12) that
0 Σ˜−11 U˜1∗M U2
−V˜2∗N−1V1Σ−11 0
≤max{ 1 σ2r, 1
˜
σ2s}N−12E∗M12
= max{ 1 σ2r, 1
˜ σ2s}
M12EN−12
= max{ 1 σ2r, 1
˜
σ2s} kEk(M N), which together with (3.9), (3.10) and Lemma 2.1(2) deduces (3.1).
Remark 3.2. If we take M =N=I, then Theorem 3.1 reduces to kB†−A†k ≤(kA†k2kB†k2+ max{kA†k22,kB†k22})kEk, (3.13)
which is the result of Theorem 3.1 in [3].
For theQ(N M)-norm we provide the following bound.
Theorem 3.3. Let A∈Cm×nr andB=A+E∈Cm×ns . Then kB†M N−A†M NkQ(N M)
(3.14)
≤ q
kA†M Nk4N M +kBM N† k4N M+kA†M Nk2N MkBM N† k2N MkEkQ(M N).
Proof. The bound (3.14) follows immediately from Lemma 2.2, (3.4)–(3.6) and (3.8).
Remark 3.4. If we take M =N=I, then Theorem 3.3 reduces to kB†−A†kQ ≤
q
kA†k42+kB†k42+kA†k22kB†k22kEkQ, which is the result of Theorem 3.2 in [3].
Theorem 3.5. Let A∈Cm×nr andB=A+E∈Cm×ns . Then
kBM N† −A†M NkF(N M)≤max{kA†M Nk2N M,kB†M Nk2N M}kEkF(M N). (3.15)
Proof. It follows from (3.3) that
U1∗MU˜1Σ˜1−Σ1V1∗N−1V˜1=U1∗M EN−1V˜1, (3.16)
U˜2∗M U1Σ1=−U˜2∗M EN−1V1, U2∗MU˜1Σ˜1=U2∗M EN−1V˜1, (3.17)
Σ˜1V˜1∗N−1V2= ˜U1∗M EN−1V2, Σ1V1∗N−1V˜2=−U1∗M EN−1V˜2. (3.18)
By (3.16), we have
Σ−11 U1∗MU˜1−V1∗N−1V˜1Σ˜−11 = Σ−11 U1∗M EN−1V˜1Σ˜−11 . (3.19)
It is easy to see that
kBM N† −A†M NkF(N M)=kN12(N−1V˜1Σ˜−11 U˜1∗M−N−1V1Σ−11 U1∗M)M−12kF
=kV∗N−12(N−12V˜1Σ˜−11 U˜1∗M12 −N−12V1Σ−11 U1∗M12)M12U˜kF
=kV∗(N−1V˜1Σ˜−11 U˜1∗M−N−1V1Σ−11 U1∗M) ˜UkF
=
V1∗ V2∗
(N−1V˜1Σ˜−11 U˜1∗M−N−1V1Σ−11 U1∗M)( ˜U1,U˜2)
F
=
V1∗N−1V˜1Σ˜−11 −Σ−11 U1∗MU˜1 −Σ−11 U1∗MU˜2
V2∗N−1V˜1Σ˜−11 0
F
(3.20) .
It follows from (3.8) and (3.20) that
2kBM N† −A†M Nk2F(N M)=kΣ˜−11 U˜1∗M U1−V˜1∗N−1V1Σ−11 k2F +kΣ˜−11 U˜1∗M U2k2F
+kV˜2∗N−1V1Σ−11 k2F+kV1∗N−1V˜1Σ˜−11 −Σ−11 U1∗MU˜1k2F
+kΣ−11 U1∗MU˜2k2F+kV2∗N−1V˜1Σ˜−11 k2F, which together with Lemma 2.2, (3.4)–(3.6) and (3.16)–(3.19) yields
2kBM N† −A†M Nk2F(N M)=kΣ˜−11 U˜1∗M EN−1V1Σ−11 k2F+kΣ˜−21 V˜1∗N−1E∗M U2k2F
+kΣ−11 U1∗M EN−1V˜1Σ˜−11 k2F+kV˜2∗N−1E∗M U1Σ−21 k2F
+kΣ−21 V1∗N−1E∗MU˜2k2F +kV2∗N−1E∗MU˜1Σ˜−21 k2F
(3.21)
≤ 1
σ2rσ˜s2(kU˜1∗M EN−1V1k2F+kU1∗M EN−1V˜1k2F) + 1
σ4r(kV˜2∗N−1E∗M U1k2F+kV1∗N−1E∗MU˜2k2F)
≤max{ 1 σ4r, 1
˜
σ4s}(kU˜1∗M EN−1V1k2F+kU1∗M EN−1V˜1k2F) +kV˜1∗N−1E∗M U2k2F+kV2∗N−1E∗MU˜1k2F
+kV˜2∗N−1E∗M U1k2F+kV1∗N−1E∗MU˜2k2F
≤2 max{ 1 σ4r, 1
˜
σ4s}kM12EN−12k2F
= 2 max{kA†M Nk4N M,kBM N† k4N M}kEk2F(M N). Therefore,
kBM N† −A†M NkF(N M)≤max{kA†M Nk2N M,kB†M Nk2N M}kEkF(M N), which implies (3.15) holds.
Remark 3.6. If we take M =N=I, then Theorem 3.5 reduces to kB†−A†kF ≤max{kA†k22,kB†k22}kEkF,
which is the result of Theorem 2.1 in [5].
Now we consider the case that rank(A) =rank(B), i.e.,A, B∈Cm×nr . Theorem 3.7. Let A, B=A+E∈Cm×nr . Then
kBM N† −A†M Nk(N M)≤[kA†M NkN MkBM N† kN M+ (kA†M NkN M+kB†M NkN M) min{kA†M NkN M,kBM N† kN M}]kEk(M N).
Proof. Let
D1=
Σ1 0 0 σrI
, D2=
Σ˜1 0 0 σ˜rI
, and
X =
Σ˜−11 U˜1∗M U1−V˜1∗N−1V1Σ−11 Σ˜−11 U˜1∗M U2
−V˜2∗N−1V1Σ−11 0
.
Then
XD1+D2X =
Σ˜−11 U˜1∗M U1Σ1−V˜1∗N−1V1 σrΣ˜−11 U˜1∗M U2
−V˜2∗N−1V1 0
+
U˜1∗M U1−Σ˜1V˜1∗N−1V1Σ−11 U˜1∗M U2
−σ˜rV˜2∗N−1V1Σ−11 0 (3.22) .
From (3.4) it is easy to see that
kΣ˜−11 U˜1∗M U1Σ1−V˜1∗N−1V1k ≤ 1
˜
σrkM12EN−12k ≤ 1
˜
σrkEk(M N), (3.23)
kU˜1∗M U1−Σ˜1V˜1∗N−1V1Σ−11 k ≤ 1
σrkM12EN−12k ≤ 1
σrkEk(M N). (3.24)
SinceUe∗M U is unitary, by Lemma 2.4 we have kUe1∗M U2k=kUe2∗M U1k. By (3.3) and (3.5), one may deduce that
Ue2∗M U1=−Ue2∗M EN−1V1Σ−11
and
Ue1∗M U2=Σe−11 (U2∗M EN−1Ve1)∗. This in turn implies that
kUe2∗M U1k ≤ 1
σrkEkM N, and
kUe1∗M U2k ≤ 1 f
σrkEkM N, respectively. Then
kU˜1∗M U2k ≤ 1
max{σr,σ˜r}kEk(M N), and thus
kσrΣ˜−11 U˜1∗M U2k ≤ σr
˜
σrkU˜1∗M U2k ≤ σr
˜
σrmax{σr,σ˜r}kEk(M N). By an analogous argument, we have
kV˜2∗N−1V1k ≤ 1
max{σr,σ˜r}kEk(M N)
and
kσ˜rV˜2∗N−1V1Σ−11 k ≤ σ˜r
σrkV˜2∗N−1V1k ≤ σ˜r
σrmax{σr,˜σr}kEk(M N), which together with (3.8), (3.22)–(3.24) and Lemma 2.3 give the desired result.
Remark 3.8. If we take M =N=I, then Theorem 3.7 reduces to kB†−A†k ≤[kA†k2kB†k2+ (kA†k2+kB†k2) min{kA†k2,kB†k2}]kEk, (3.25)
which is the result of Theorem 3.3 in [3].
For the weightedF-norm we have:
Theorem 3.9. Let A, B=A+E∈Cm×nr . Then
kB†M N−A†M NkF(N M)≤ kA†M NkN MkBM N† kN MkEkF(M N). (3.26)
Proof. Since in (1.1)U∗MU ,˜ V N˜ −1V andV∗N−1V˜ are unitary, by Lemma 2.4 we have
kU˜1∗M U2kF =kU˜2∗M U1kF,kU2∗MU˜1kF =kU1∗MU˜2kF, (3.27)
kV˜1∗N−1V2kF =kV˜2∗N−1V1kF,kV2∗N−1V˜1kF =kV1∗N−1V˜2kF. (3.28)
It follows from (3.17), (3.18) and (3.27), (3.28) that
2kBM N† −A†M Nk2F(N M)=kΣ˜−11 U˜1∗M EN−1V1Σ−11 k2F+kΣ−11 U1∗M EN−1V˜1Σ˜−11 k2F
+ 1
˜
σr2(kU˜1∗M U2k2F+kV2∗N−1V˜1k2F) + 1
σr2(kV˜2∗N−1V1k2F+kU1∗MU˜2k2F)
≤ 1
˜
σr2σ2r(kU˜1∗M EN−1V1k2F +kU1∗M EN−1V˜1k2F) + 1
˜
σr2(kU˜2∗M U1k2F+kV1∗N−1V˜2k2F) + 1
σr2(kV˜1∗N−1V2k2F+kU2∗MU˜1k2F)
= 1
˜
σr2σ2r(kU˜1∗M EN−1V1k2F +kU1∗M EN−1V˜1k2F) + 1
˜
σr2(kU˜2∗M EN−1V1Σ−11 k2F+kΣ−11 U1∗M EN−1V˜2k2F) + 1
σr2(kΣ˜−11 U˜1∗M EN−1V2k2F+kU2∗M EN−1V˜1Σ˜−11 k2F)
≤ 2
˜
σr2σ2rkM12EN−12k2F = 2kA†M Nk2N MkB†M Nk2N MkEk2F(M N), which implies (3.26) holds.
Remark 3.10. Since
kA†M NkN MkBM N† kN M ≤max{kA†M Nk2N M,kBM N† k2N M}, the bound is sharper than the one in (3.15).
For the weightedQ-norm we have:
Theorem 3.11. LetA, B =A+E∈Cm×nr . Then kB†M N−A†M NkQ(N M)≤√
3kA†M NkN MkBM N† kN MkEkQ(M N). (3.29)
Proof. By (3.20) we have
kBM N† −A†M NkQ(N M)=
V1∗N−1V˜1Σ˜−11 −Σ−11 U1∗MU˜1 −Σ−11 U1∗MU˜2
V2∗N−1V˜1Σ˜−11 0
Q.
It follows from (3.8), (3.27), (3.28) and Lemma 2.2 that
2kBM N† −A†M Nk2Q(N M)≤ kΣ˜−11 U˜1∗M U1−V˜1∗N−1V1Σ−11 k2Q+kΣ˜−11 U˜1∗M U2k2Q
+kV˜2∗N−1V1Σ−11 k2Q+kV1∗N−1V˜1Σ˜−11 −Σ−11 U1∗MU˜1k2Q
+kΣ−11 U1∗MU˜2k2Q+kV2∗N−1V˜1Σ˜−11 k2Q, which together with (3.4)–(3.7) and (3.16)–(3.19) gives that
2kBM N† −A†M Nk2Q(N M)≤ kΣ˜−11 U˜1∗M EN−1V1Σ−11 k2Q+kΣ−11 U1∗M EN−1V˜1Σ˜−11 k2Q
+ 1
˜
σ2r(kU˜1∗M U2k2Q+kV2∗N−1V˜1k2Q) + 1
σ2r(kV˜2∗N−1V1k2Q+kU1∗MU˜2k2Q)
≤ 1
σ2rσ˜r2(kU˜1∗M EN−1V1k2Q+kU1∗M EN−1V˜1k2Q) + 1
˜
σ2r(kU˜2∗M U1k2Q+kV1∗N−1V˜2k2Q) + 1
σ2r(kV˜1∗N−1V2k2Q+kU2∗MU˜1k2Q)
= 1
σ2rσ˜r2(kU˜1∗M EN−1V1k2Q+kU1∗M EN−1V˜1k2Q) + 1
˜
σ2r(kU˜2∗M EN−1V1Σ−11 k2Q+kΣ−11 U1∗M EN−1V˜2k2Q) + 1
σ2r(kΣ˜−11 U˜1∗M EN−1V2k2Q+kU2∗M EN−1V˜1Σ˜−11 k2Q)
≤ 1
σ2rσ˜r2(kU˜1∗M EN−1V1k2Q+kU˜2∗M EN−1V1k2Q
+kU˜1∗M EN−1V2k2Q+kU1∗M EN−1V˜1k2Q
+kU1∗M EN−1V˜2k2Q+kU2∗M EN−1V˜1k2Q)
≤6kA†M Nk2N MkB†M Nk2N MkEk2Q(M N), which implies that the inequality (3.29) holds.
Remark 3.12. If we take M =N =I, then Theorem 3.11 reduces to kB†−A†kQ≤√
3kA†k2kB†k2kEkQ, which is the result of Theorem 3.4 in [3].
It is easy to see that
√3kA†M NkN MkBM N† kN M ≤ q
kA†M Nk4N M+kBM N† k4N M+kA†M Nk2N MkB†M Nk2N M, which implies that the bound in (3.29) is sharper than the one in (3.14).
4. Multiplicative perturbation bounds. In this section, we present optimal multiplicative perturbation bounds of the weighted Moore-Penrose inverse. Let B be a multiplicative perturbed matrix ofA, i.e., B =D1AD2, where D1 and D2 are m×mand n×nnonsingular matrices, then rank(A) =rank(B).
Theorem 4.1. Let A∈Cm×n andB =D1∗AD2, where D1 andD2 are respec- tively m×m andn×nnonsingular matrices. Then
kBM N† −A†M Nk(N M)≤max{kA†M NkN M,kBM N† kN M}Ψ1(D1, D2), (4.1)
where
Ψ1(D1, D2) =kIm−D∗1k(M M)+kIm−D−∗1 k(M M)+kIn−D2k(N N)+kIn−D2−1k(N N).
Proof. LetA andB have the MN-SVDs (3.2). Clearly we have B−A=B(In−D2−1) + (D∗1−Im)A
= (Im−D1−∗)B+A(D2−In).
(4.2)
It follows from (3.3) and (4.2) that
U˜1Σ˜1V˜1∗−U1Σ1V1∗= ˜U1Σ˜1V˜1∗(In−D2−1) + (D1∗−Im)U1Σ1V1∗
= (Im−D1−∗) ˜U1Σ˜1V˜1∗+U1Σ1V1∗(D2−In).
(4.3)
By (4.3) we obtain
Σ˜1V˜1∗N−1V1−U˜1∗M U1Σ1= ˜Σ1V˜1∗(In−D2−1)N−1V1+ ˜U1∗M(D∗1−Im)U1Σ1,
U1∗MU˜1Σ˜1−Σ1V1∗N−1V˜1=U1∗M(Im−D1−∗) ˜U1Σ˜1+ Σ1V1∗(D2−In)N−1V˜1,
U2∗MU˜1Σ˜1=U2∗M(Im−D−∗1 ) ˜U1Σ˜1, U˜2∗M U1Σ1=−U˜2∗M(D1∗−Im)U1Σ1, Σ˜1V˜1∗N−1V2= ˜Σ1V˜1∗(In−D2−1)N−1V2,
Σ1V1∗N−1V˜2=−Σ1V1∗(D2−In)N−1V˜2, from which one may deduce that
V˜1∗N−1V1Σ−11 −Σ˜−11 U˜1∗M U1
(4.4)
= ˜V1∗(In−D−12 )N−1V1Σ−11 + ˜Σ−11 U˜1∗M(D1∗−Im)U1, Σ˜−11 U˜1∗M U2= ˜Σ−11 U˜1∗(Im−D−11 )M U2,
(4.5)
−V˜2∗N−1V1Σ−11 = ˜V2∗N−1(D∗2−In)V1Σ−11 , (4.6)
Σ−11 U1∗MU˜1−V1∗N−1V˜1Σ˜−11 (4.7)
= Σ−11 U1∗M(Im−D1−∗) ˜U1+V1∗(D2−In)N−1V˜1Σ˜−11 ,
−Σ−11 U1∗MU˜2= Σ−11 U1∗(D1−Im)MU˜2, (4.8)
V2∗N−1V˜1Σ˜−11 =V2∗N−1(In−D2−∗) ˜V1Σ˜−11 . (4.9)
By (3.7) and (4.4)–(4.6) we obtain kBM N† −A†M Nk(N M)
=
L Σ˜−11 U˜1∗(Im−D1−1)M U2
V˜2∗N−1(D∗2−In)V1Σ−11 0
≤
−Σ˜−11 U˜1∗M(D∗1−Im)U1 Σ˜−11 U˜1∗(Im−D−11 )M U2
0 0
+
−V˜1∗(In−D−12 )N−1V1Σ−11 0 V˜2∗N−1(D2∗−In)V1Σ−11 0
≤
M Σ˜−11 U˜1∗M12M−12(Im−D1−1)M12M12U2
0 0
! +
−V˜1∗N−12N12(In−D2−1)N−12N−12V1Σ−11 0 V˜2∗N−12N−12(D2∗−In)N12N−12V1Σ−11 0
!, (4.10)
where
L ≡ −V˜1∗(In−D−12 )N−1V1Σ−11 −Σ˜−11 U˜1∗M(D1∗−Im)U1, M ≡ −Σ˜−11 U˜1∗M12M12(D1∗−Im)M−12M12U1. This implies (4.1) holds.
Remark 4.2. If we take M =N=I, then Theorem 4.1 reduces to kB†−A†k ≤max{kA†k2,kB†k2}Ψ(D1, D2),