some remarks on the perturbation of polar decompositions for rectangular matrices

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONSNumer. Linear Algebra Appl. 2006; 13:327–338Published online 24 August 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nla.463

Some remarks on the perturbation of polar decompositions forrectangular matrices

Wen Li1;∗;† and Weiwei Sun2;‡

1School of Mathematics Science; South China Normal University; Guangzhou; 510631;People’s Republic of China

2Department of Mathematics; City University of Hong Kong; Hong Kong; People’s Republic of China

SUMMARY

In this article we focus on perturbation bounds of unitary polar factors in polar decompositions forrectangular matrices. First we present two absolute perturbation bounds in unitarily invariant norms andin spectral norm, respectively, for any rectangular complex matrices, which improve recent results ofLi and Sun (SIAM J. Matrix Anal. Appl. 2003; 25:362–372). Secondly, a new absolute bound forcomplex matrices of full rank is given. When ‖A− A‖2 � ‖A− A‖F, our bound for complex matrices isthe same as in real case. Finally, some asymptotic bounds given by Mathias (SIAM J. Matrix Anal.Appl. 1993; 14:588–593) for both real and complex square matrices are generalized. Copyright ? 2005John Wiley & Sons, Ltd.

KEY WORDS: polar decomposition; perturbation bound; unitarily invariant norm

1. INTRODUCTION

Let Cm×n(Rm×n) be the set of m× n complex (real) matrices and Cm×nr (Rm×n

r ) be the set ofm× n complex (real) matrices having rank r. Throughout this paper we always assume thatm¿ n. We denote by ‖ · ‖2, ‖ · ‖F and ‖ · ‖ the spectral norm, the Frobenius norm and thegeneral unitarily invariant norm, respectively. Let A, A∈Cm×n

r (m¿n) with the singular value

∗Correspondence to: Wen Li, School of Mathematics Science, South China Normal University, Guangzhou, 510631,People’s Republic of China.

†E-mail: [email protected]‡E-mail: [email protected]

Contract=grant sponsor: Guangdong Provincial Natural Science Foundation; contract=grant number: 31496Contract=grant sponsor: Natural Science Foundation of Guangdong Provincial Universities; contract=grantnumber: 0119Contract=grant sponsor: Excellent Talent Foundation of Guangdong Province; contract=grant number: Q02084Contract=grant sponsor: City University of Hong Kong Research Grant; contract=grant number: 7001467

Received 26 September 2003Copyright ? 2005 John Wiley & Sons, Ltd. Revised 10 October 2004

328 W. LI AND W. SUN

decompositions

A=U�V ∗ and A= U �V∗ (1)

where

�=

(�1 0

0 0

)∈Cm×n

r and �=

(�1 0

0 0

)∈Cm×n

r

�1 =diag(�1; : : : ; �r), �1 =diag(�1; : : : ; �r); �1¿ · · ·¿�r ¿ 0 and �1¿ · · ·¿�r ¿ 0, the super-script ∗ denotes conjugate transpose. Let

H =V1�1V ∗1 ; Q=U1V ∗

1 ; H = V1�1V1∗ and Q= U1V1∗

where U =(U1; U2), U =(U1; U2)∈Cm×m and V =(V1; V2), V =(V1; V2)∈Cn×n are unitary,U1; U1 ∈Cm×r

r ; V1; V1 ∈Cn×rr . The polar decompositions of A and A are de�ned by

A=QH; A= QH (2)

The Q and Q are called the (sub) unitary polar factors of A and A. When r=m= n;U =U1and V =V1.Much e�ort has been made for estimating perturbation bounds of the unitary polar factor

Q in di�erent norms, see References [1–6]. It has been noted that there is some signi�cantdi�erence between perturbation bounds for square matrices and for non-square matrices. ForA; A(=A+ E)∈Rm×n with r=m= n, the perturbation of Q is proportional to the reciprocalof the sum of two smallest singular values and the best bound in the Frobenius norm [4] is

‖Q − Q‖F6 4�n + �n−1 + �n + �n−1

‖E‖F (3)

However, it is not true when n �= m. For non-square matrices, the condition number of theunitary factor in Frobenius norm is 1=�n [7]. The following example shows that in any unitarilyinvariant norm, the perturbation of Q could be proportional to the reciprocal of the smallestsingular value.

Example 1.1Let

Q=

⎛⎜⎜⎜⎜⎜⎝1√2

1√2

− 1√2

1√2

0 0

⎞⎟⎟⎟⎟⎟⎠ ; Q=

⎛⎜⎜⎜⎜⎜⎜⎝

1√2

�√2

− 1√2

�√2

0√1− �2

⎞⎟⎟⎟⎟⎟⎟⎠ ; H = H =diag(�1; �2)

with 06�61. Here we have

‖E‖=�2‖Q −Q‖For A, A∈Cm×n, existing perturbation bounds for square matrices and non-square matrices aredi�erent [1, 3, 4]. There are two types of perturbation bounds: absolute bound and asymptotic

Copyright ? 2005 John Wiley & Sons, Ltd. Numer. Linear Algebra Appl. 2006; 13:327–338

PERTURBATION BOUNDS FOR EIGENVALUES OF NORMAL MATRICES 329

bound. The former is a bound without any assumption on the size of the perturbation and thelatter is true only when the perturbation in certain norm is small enough. In this article, wefocus on rectangular matrices and present some new bounds in both absolute and asymptoticsenses. The rest of this article is organized as follows.

In Section 2, we present two new absolute perturbation bounds for any rectangular matrixin the spectral norm and in the unitarily invariant norm, respectively, which improve thebounds in Reference [1]. In Section 3, we present an absolute perturbation bound for complexmatrices of full rank. It is known that the perturbation of unitary polar factors is proportionalto the reciprocal of the two smallest singular values in real case and proportional to only thesmallest singular value in complex case in general. Our bound shows that for those large scaleof complex matrices, when ‖E‖2 is much small than ‖E‖F, the perturbation bound in complexcase is the same as in real case. In Section 4, we extend some asymptotic perturbation boundsgiven in Reference [8] for the unitarily invariant norm to rectangular matrices of full rank.

2. ABSOLUTE BOUNDS FOR GENERAL COMPLEX MATRICES

The absolute perturbation bound of the unitary factor Q has been studied by many authors,e.g. see References [1, 3, 4]. The bounds obtained by Li are

‖Q − Q‖6 2�n + �n

‖E‖ (4)

for r=m= n and

‖Q − Q‖6(

2�n + �n

+1

max{�n; �n})

‖E‖ (5)

for r= n¡m. For more general case, A, A∈Cm×nr , the authors Li and Sun [1] provided the

bound

‖Q − Q‖6(

2�r + �r

+2

max{�r; �r})

‖E‖ (6)

for the unitarily invariant norm and

‖Q − Q‖26√(

2�r + �r

)2+

2max{�2r ; �2r }

‖E‖2 (7)

for the spectral norm.Two new perturbation bounds are given in the following two theorems, respectively.

Theorem 2.1Let A and A∈Cm×n

r (n¡m) have the SVDs in (1). Then

‖Q − Q‖6(

2�r + �r

+1

min{�r; �r})

‖E‖ (8)



Theorem 2.2Let A and A∈Cm×n

r (n¡m) have the SVDs in (1). Then

‖Q − Q‖226�‖E‖22 (9)

where

�=12

⎡⎢⎣( 2�r + �r

)2+1�2r+1�2r+

√√√√((2

�r + �r

)2+1�2r+1�2r

)2− 4�2r �

2r

⎤⎥⎦ (10)

Remark 2.1It is easy to show that bound (9) is sharper than bound (7). Since A is perturbed to A, ‖E‖may be very small and �r ≈ �. Asymptotically, bound (8) improves bound (6) by a factor 1.5and

�≈ 2:618�2r

¡3�2r

≈(

2�r + �r

)2+

2max{�2r ; �2r }

Proof of Theorem 2.1Let E=A− A. By the block forms of U and V given in Section 1, we see that

U ∗EV =

(U ∗1 EV1 U ∗

1 EV2

U ∗2 EV1 U ∗

2 EV2

)

and hence (e.g. see Theorem 3.7 of Reference [9]) for any unitarily invariant norm

‖E‖¿∥∥∥∥∥(

0 U ∗1 EV2

U ∗2 EV1 0

)∥∥∥∥∥ (11)

Since E=A− A=U1�1V ∗1 − U1�1V1∗, we have

�1V ∗1 V2 =U

∗1 EV2 (12)

and

−U ∗2 U1

˜�1 =U ∗2 EV1 (13)

It follows that

‖E‖¿∥∥∥∥∥(

0 �1V ∗1 V2

−U ∗2 U1�1 0

)∥∥∥∥∥ (14)

Let

B=

(0 �1V ∗

1 V2

−U ∗2 U1�1 0

)≡(0 B1

B2 0

)



and

B=

(0 V ∗

1 V2

−U ∗2 U1 0

)≡(0 B1

B2 0

)Then

BB∗=

(B1B∗

1 0

0 B2B∗2

)

We arrange the eigenvalues of B1B1∗ and B2∗B2 in an increasing order. By Ostrowski Theorem(e.g. see Theorem 4.5.9 of Reference [10]) we have

�k(B1B∗1 )¿ �2r �k((V

∗1 V2)(V

∗1 V2)

∗)

¿min{�2r ; �2r }�k(B1B1∗)

and

�k(B2B∗2 ) = �k(�(U

∗2 U1)

∗(U ∗2 U1)�)

¿ �2r �k((U∗2 U1)

∗(U ∗2 U1))

¿min{�2r ; �2r }�k(B2∗B2)

where �k(∗) denotes the kth eigenvalue of ∗. We arrange the singular values of B and B ina decreasing order. From the above spectral inequalities one may easily deduce that

�i(B)¿min{�r; �r}�i(B)By Theorem 3.3 of Reference [9], we have ‖B‖¿min{�r; �r}‖B‖, and by (14)

‖E‖¿min{�r; �r}‖B‖ (15)

Since

U ∗(Q −Q)V =U ∗(U1V1 −U1V1)V =(U ∗1 U1 − V ∗

1 V1 −V ∗1 V2

U ∗2 U1 0

)(16)

we have

‖Q −Q‖=∥∥∥∥∥(U ∗1 U1 − V ∗

1 V1 −V ∗1 V2

U ∗2 U1 0

)∥∥∥∥∥and by (15)

‖Q −Q‖6∥∥∥∥∥(U ∗1 U1 − V ∗

1 V1 0

0 0

)∥∥∥∥∥+ ‖B‖

6 ‖U ∗1 U1 − V ∗

1 V1‖+1

min{�r; �r}‖E‖



By Lemma 3.3 in Reference [1],

‖U ∗1 U1 − V ∗

1 V1‖62

�r + �r‖E‖ (17)

Bound (8) follows immediately.

Proof of Theorem 2.2By (16), we have

U ∗(Q −Q)(Q −Q)∗U

=

((U ∗

1 U1 − V ∗1 V1)(U

∗1 U1 − V ∗

1 V1)∗ + (V ∗

1 V2)(V∗1 V2)

∗ (U ∗1 U1 − V ∗

1 V1)(U1∗U2)∗

(U1∗U2)(U ∗1 U1 − V ∗

1 V1)∗ (U1∗U2)(U1∗U2)∗

)(18)

and moreover,

‖Q −Q‖226�(‖U ∗

1 U1 − V ∗1 V1‖22 + ‖V ∗

1 V2‖22 ‖U ∗1 U1 − V ∗

1 V1‖2‖U1∗U2‖2‖U1∗U2‖2‖U ∗

1 U1 − V ∗1 V1‖2 ‖U1∗U2‖22

)

where �(·) denotes the spectral radius. By (11)–(13),�r‖U1∗U2‖26‖E‖2; �r‖V ∗

1 V2‖26‖E‖2which together with (17) gives

‖Q −Q‖226�‖E‖22where

�=�

⎛⎜⎜⎜⎝(

2�r + �r

)2+1�2r

(2

�r + �r

)1�r(

2�r + �r

)1�r

1�2r

⎞⎟⎟⎟⎠Equation (9) can be obtained by a simple calculation.

3. ABSOLUTE BOUNDS FOR FULL RANK COMPLEX MATRICES

The following inequality was given by authors in Reference [4] for A, A∈Cn×nn .[(1− �)

(�n−1 + �n−1

2

)+ �

(�n + �n2

)]‖Q −Q‖F6‖E‖F (19)

where �=(‖Q−Q‖22)=(‖Q−Q‖2F). Based on this inequality, (3) is obtained for real and squarematrices. In this section, �rst we extend (19) to any rectangular matrices and then, provide anew absolute perturbation bound for complex matrices in terms of this inequality.



Here we use the same notation as in References [1, 4]. Let Ip be the p×p identity matrixand

I (p)m;n ≡(Ip 0

0 0

)

For simplicity we replace I (p)m;n with I (p). Let A, A∈Cm×nr have the SVD in (1) and let S= U∗U

and T = V∗V have the block form

S=

(S11 S12

S21 S22

)∈Cm×m and T =

(T11 T12

T21 T22

)∈Cn×n

where both S11 and T11 are r × r. Then S and T are unitary matrices. LetM = 2I − S∗

11T11 − T ∗11S11; M =2I − T11S∗

11 − S11T ∗11

W = I (r) − S∗I (r)T; W = I (r) − SI (r)T ∗ (20)

and let mij and mij denote the (i; j) entry of M and M , respectively.The following two inequalities can be found in References [1, 3, 4].

‖Q − Q‖2F = ‖SI (r) − I (r)T‖2F = ‖W‖2F = ‖W‖2F = tr(M)= tr(M) (21)

‖E‖2F¿ �[(�r−1 + �r−1 − �) tr(M)− (�r−1 − �r)mrr − (�r−1 − �r)mrr] (22)

for A, A∈Cn×nn . It is easy to show that these two inequalities hold for A, A∈Cm×nr by an

analogous approach.

Lemma 3.1Let A, A∈Cm×n

n have the SVDs in (1). Then

‖M‖2 = ‖M‖2 = ‖Q −Q‖22ProofBy (20), M =TMT ∗, which implies that for any unitarily invariant norm ‖ · ‖, ‖M‖= ‖M‖.Since

W =

(I − S∗

11T

−S∗12T

)we have

W ∗W =2I − S∗11T − T ∗S11 =M

from which one may deduce that

‖M‖2 = ‖M‖2 = ‖W ∗W‖2 = ‖W‖22 = ‖Q −Q‖22which proves our assertion.



Lemma 3.2Let A, A∈Cm×n

n have the SVDs in (1). Then inequality (19) holds.

ProofBy (21), (22) and Lemma 3.1,

‖E‖2F¿ �[(�n−1 + �n−1 − �) tr(M)− (�n−1 − �n)mnn − (�n−1 − �n)mnn]¿ �[(�n−1 + �n−1 − �) tr(M)− (�n−1 − �n + �n−1 − �n)max{‖M‖2; ‖M‖2}]= �[(�n−1 + �n−1 − �)‖Q − Q‖2F − (�n−1 − �n + �n−1 − �n)‖Q −Q‖22]

where we have noted the fact mii6‖M‖2 and mii6‖M‖2. By maximizing the right-hand sideof the above inequality over � we have[

(1− �)(�n−1 + �n−1

2

)+ �

(�n + �n2

)]‖Q −Q‖F6‖E‖F

with �=(‖Q −Q‖22)=(‖Q −Q‖2F), which proves the lemma.Theorem 3.3Let A, A∈Cm×n

n have the SVDs in (1). Then at least one of the following bounds holds

‖Q −Q‖F¡√2�‖E‖2 (23)

and

‖Q −Q‖F6 4�n + �n + �n−1 + �n−1

‖E‖F (24)

where � is de�ned in (10).

ProofAssume that (23) is not true, i.e.

‖Q −Q‖F¿√2�‖E‖2

By (10),

‖Q − Q‖26√�‖E‖26

√22 ‖Q − Q‖F

Hence

�=‖Q −Q‖22‖Q −Q‖2F

612

which with Lemma 3.2 leads to (24).

Remark 3.1In (23) and (24), the Frobenius norm of perturbation of the unitary polar factor is boundedin two di�erent norms of perturbation of A, spectral norm and Frobenius norm. Neither of



these two bounds is sharper than other one in general. However, for large scale matrices, weoften have ‖E‖2 � ‖E‖F and in this case, (24) always holds.

4. ASYMPTOTIC BOUNDS ON UNITARILY INVARIANT NORMS

Mathias in Reference [8] provided the asymptotic bounds

‖Q −Q‖6 2‖E‖&E&2

× log(1− &E&2

�n + �n−1

)(25)

‖Q −Q‖6 max06t61

(2

�n(t) + �n−1(t)

)‖E‖ (26)

for A, A∈Rn×nn and the bounds

‖Q −Q‖6− ‖E‖‖E‖2 log

(1− ‖E‖2

�n

)(27)

‖Q −Q‖6 max06t61

{1�n(t)

}‖E‖ (28)

for A, A∈Cn×nn , where &A&k =∑k

i=1 �i(A) de�nes the Ky Fan k-norm. Here we extendthese bounds to matrices of full rank. First we consider the case that A; A∈Rm×n

n ; m¿n. LetA(t)=A+ tE, t ∈ [0; 1], and ‖E‖2¡�n and let

A(t)=U (t)�(t)V (t)∗ (29)

be the SVD of A(t), where �(t)=diag(�1(t); : : : ; �n(t)) with �1(t)¿ · · ·¿�n(t). Then for anyt ∈ [0; 1]

�n(t)¿�n − t‖E‖2¿ 0

Hence A(t) has full rank for any t ∈ [0; 1]. This implies that A(t) has the unique polar de-composition

A(t)=Q(t)H (t)

Di�erentiating both sides of the above equation gives

E dt=dQ(t)H (t)+Q(t) dH (t) (30)

Since Q(t)∗Q(t)= I , we have

dQ(t)∗Q(t) +Q(t)∗dQ(t)=0

and moreover,

[Q(t)∗E−E∗Q(t)] dt=Q(t)∗dQ(t)H (t)−H (t) dQ(t)∗Q(t)



which leads to

[Q(t)∗E − E∗Q(t)] dt=Q(t)∗dQ(t)H (t) +H (t)Q(t)∗dQ(t) (31)

From (29) one may deduce that

H (t)=V (t)�(t)V (t)∗

Let X (t)=V (t)∗Q(t)∗dQ(t)V (t) and Y (t)=V (t)∗Q(t)∗EV (t) dt. Then Equation (31) can berewritten by

X (t)�(t) + �(t)X (t)=Y (t)− Y (t)∗ (32)

or equivalently,

X (t) = C ◦ (Y (t)− Y (t)∗)where C=(1=(�i(t) + �j(t))). It follows from Lemma 2.1 of Reference [8] that

&X (t)&k 61

�n(t) + �n−1(t)&Y (t)− Y (t)∗&k

62

�n(t) + �n−1(t)&Y (t)&k

and therefore,

&Q(t)∗dQ(t)&k62

�n(t) + �n−1(t)&E&k dt (33)

Let (Q(t); Q(t)) be unitary. Multiplying Q(t)∗ on the both sides of (30) gives

Q(t)∗Edt= Q(t)∗ dQ(t)H (t)

and moreover,

Q(t)∗ dQ(t)= Q(t)∗EH (t)−1 dt

Hence,

&Q(t)∗ dQ(t)&k =1�n(t)

‖E&k dt (34)

From (34) and (35) we have

&dQ(t)&k 6&Q(t)∗dQ(t)&k + &Q(t)

∗dQ(t)&k

6(

2�n(t) + �n−1(t)

+1�n(t)

)&E&k dt



We see that

&Q −Q&k = &Q(1)−Q(0)&k =∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∫ 1

0dQ(t)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣k

6∫ 1

0&dQ(t)&k

6&E&k∫ 1

0

(2

�n(t) + �n−1(t)+

1�n(t)

)dt

6&E&k∫ 1

0

(2

�n + �n−1 − t&E&2+

1�n − t‖E‖2

)dt

6−[

2

&E&2log

(1− &E&2

�n + �n−1

)+

1‖E‖2 log

(1− ‖E‖2

�n

)]&E&k

Two asymptotic perturbation bounds for A, A∈Rm×nn , are given in the following theorem. The

proof can be obtained immediately by the last equation and Lemma 1.1 of Reference [8].

Theorem 4.1Let A, A∈Rm×n

n , m¿n and ‖E‖2¡�n. Then for any unitarily invariant norm ‖ · ‖,

‖Q −Q‖6−[

2

&E&2log

(1− &E&2

�n + �n−1

)+

1‖E‖2 log

(1− ‖E‖2

�n

)]‖E‖

and

‖Q −Q‖6 max06t61

(2

�n(t) + �n−1(t)+

1�n(t)

)‖E‖

Two asymptotic bounds in complex case can be obtained analogously.

Theorem 4.2Let A, A∈Cm×n

n , m¿n and ‖E‖2¡�n. Then for any unitarily invariant norm ‖ · ‖,

‖Q −Q‖6− 2‖E‖‖E‖2 log

(1− ‖E‖2

�n

)and

‖Q −Q‖6 max06t61

{2�n(t)

}‖E‖

ACKNOWLEDGEMENTS

The authors would like to thank the referees for their valuable comments.



REFERENCES

1. Li W, Sun W. New perturbation bounds for unitary polar factors. SIAM Journal on Matrix Analysis andApplications 2003; 25:362–372.

2. Li RC. A perturbation bound for the generalized polar decomposition. BIT Numerical Mathematics 1993;33:304–308.

3. Li RC. New perturbation bounds for the unitary polar factor. SIAM Journal on Matrix Analysis andApplications 1995; 16:327–332.

4. Li W, Sun W. Perturbation bounds for unitary and subunitary polar factors. SIAM Journal on Matrix Analysisand Applications 2002; 23:1183–1193.

5. Barrlund A. Perturbation bounds on the polar decomposition. BIT Numerical Mathematics 1989; 30:101–103.6. Chen XS, Li W, Sun W. Some new perturbation bounds for the generalized polar decomposition. BIT NumericalMathematics 2004; 44:237–244.

7. Chatelin F, Gratton S. On the condition numbers associated with the polar factorization of a matrix. NumericalLinear Algebra with Applications 2000; 7:337–354.

8. Mathias R. Perturbation bounds for the polar decomposition. SIAM Journal on Matrix Analysis andApplications 1993; 14:588–593.

9. Sun JG. Matrix Perturbation Analysis. Science Press: Beijing, 2001.10. Horn RA, Johnson CR. Matrix Analysis. Cambridge University Press: Cambridge, MA, 1985.


some remarks on the perturbation of polar decompositions for rectangular matrices

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