Relative Perturbation Bounds for the Unitary Polar Factor Ren-Cang Li Department of Mathematics University of California at Berkeley Berkeley, California 9470 èli@math.berkeley.eduè July 5, 994 Computer Science Division Technical Report UCBèèCSD-94-854, University of California, Berkeley, CA 9470, December, 994. Abstract Let B be an m n èm nè complex matrix. It is known that there is a unique polar decomposition B = QH, where Q Q = I, the n n identity matrix, and H is positive denite, provided B has full column rank. This paper addresses the following question: how much may Q change if B is perturbed to e B = D BD? Here D and D are two nonsingular matrices and close to the identities of suitable dimensions. Known perturbation bounds for complex matrices indicate that in the worst case, the change in Q is proportional to the reciprocal of the smallest singular value of B. In this paper, we will prove that for the above mentioned perturbations to B, the change in Q is bounded only by the distances from D and D to identities! As an application, we will consider perturbations for one-side scaling, i.e., the case when G = D B is perturbed to e G = D e B, where D is usually a nonsingular diagonal scaling matrix but for our purpose we do not have to assume this, and B and e B are nonsingular. This material is based in part upon work supported by Argonne National Laboratory under grant No. 05540 and the University of Tennessee through the Advanced Research Projects Agency under contract No. DAAL03-9-C-0047, by the National Science Foundation under grant No. ASC-9005933, and by the National Science Infrastructure grants No. CDA-87788 and CDA-94056.
Let B be an m n èm nè complex matrix. It is known that there are Q with orthonormal column vectors, i.e., Q Q = I, and a unique positive semidenite H such that B = QH: èè Hereafter I denotes an identity matrix with appropriate dimensions which should be clear from the context or specied. The decomposition èè is called the polar decomposition of B. If, in addition, B has full column rank then Q is uniquely determined also. In fact, H =èb Bè = ; Q = BèB Bè,= ; èè where superscript ë " denotes conjugate transpose. The decomposition èè can also be computed from the singular value decomposition èsvdè B = UV by H = V V ; Q = U V ; è3è è! where U =èu ;U è and V are unitary, U is m n, = and 0 = diag è ;:::; n è is nonnegative. There are many published bounds upon how much the two factor matrices Q and H may change if entries of B are perturbed in arbitrary manner ë,, 3,4,6,5,7,8,9ë. In these papers, no assumption was made on how B was perturbed unlike what we are going to do here. In this paper, we obtain some bounds for the perturbations of Q, assuming B is complex and is perturbed to B e = D BD, where D and D are two nonsingular matrices and close to the identities of suitable dimensions. Assume also B has full column rank and so do B e = D BD. Let B = QH; e B = e Q e H è4è be the polar decompositions of B and e B respectively, and let B = UV ; e B = e U e e V è5è be the SVDs of B and B, e respectively, where U e =èu e ; U e è, U e! is m n, and e e =è and e = diag èe 0 ;:::;e n è. Assume as usual that n é 0; and e e n é 0: è6è
It follows from èè and è5è that Q = U V ; e Q = e U e V : In what follows, kxk F denotes the Frobenius norm which is the square root of the trace of X X. Then eu è e B, BèV = e e V V, e U U; eu è e B, BèV = e U èd BD, D B + D B, BèV and similarly = e U h ebèi, D, è+èd, IèB i V = e e V èi, D, èv + e U èd, IèU; U è e B, Bè e V = U e U e, V e V; U è e B, Bè e V = U èd BD, BD + BD, Bèe V = U hèi, D, è e B + BèD, Iè i e V = U èi, D, è e U e +V èd, Ièe V: Therefore, we obtained two perturbation equations. ee V V, U e U = e V e èi, D, èv + U e èd, IèU; è7è U e U e, V V e = U èi, D, è e U+V e èd, Ièe V: è8è The rst n rows of the equation è7è yields e e V V, e U U = e e V èi, D, èv + e U èd, IèU : è9è The rst n rows of the equation è8è yields U e U e, V e V = U èi, D, è e U e + V èd, Ièe V; on taking conjugate transpose of which, one has e e U U, e V V = e e U èi, D, èu + e V èd, IèV : è0è Now subtracting è0è from è9è leads to e è e U U, e V V è+èe U U, e V V è èè = e h eu èi, D, èu, e V èi, D, èv i + h ev èd, IèV, e U èd, IèU i : 3
Set X = e U U, e V V =èx ij è; èè E = e U èi, D, èu, e V èi, D, èv =èe ijè; è3è ee = e V èd, IèV, e U èd, IèU =èe ij è: è4è Then the equation èè reads e X+X = e E+ e E, or componentwisely, e i x ij + x ij j = e i e ij + e ij j.thus q q jèe i + j èx ij j e + i je j ij j + je ij j è jx ij j e i + j èe i + j è èje ijj + je ij j è je ij j + je ij j : Summing on i and j for i; j =; ; ;n produces Notice that and Lemma kxk F = nx i; j= jx ij j kek F + k e Ek F : X = e U U, e V V = e V èe V e U U V, IèV = e V è e Q Q, IèV; è kxk F = k e Q Q, Ik F ; k e Q Q, Ik F kek F ki, D, k F + ki, D, k F; k e EkF kd, Ik F + kd, Ik F : è5è r ki, D, k F + ki, D, k F +èkd, Ik F + kd, Ik Fè : When m = n, both Q and e Q are unitary. Thus k e Q Q, Ik F = kq, e QkF, and Lemma yields Theorem Let B and B e = D BD be two n n nonsingular complex matrices whose polar decompositions are given by è4è. Then kq, e QkF r ki, D, k F + ki, D, k F +èkd, Ik F + kd, Ik F è è6è p q ki, D, k + ki, D, F k F + kd, Ik F + kd, Ik F : 4
If, however, m é n, then it follows from the last m, n rows of the equations è7è and è8è that eu U = e U èd, IèU and U e U e = U èi, D, è e U e : Since we assume that both B and e B have full column rank, both and e are nonsingular diagonal matrices. So Therefore, we have eu U = U e èd, IèU and U e U = U èi, D, è e U : k e U U k F kd, Ik F and ku e U k F = ki, D, k F: è7è Notice that èu V ;U è=èq; U è and èe UV e ; U e è=èe Q; U e è are unitary. Hence U Q =0= U e e Q and è! kq, QkF e = kèq; U è èq, QèkF e I, Q e = Q,U e Q F q ki, Q e Qk F + k,u e U V e k F q ki, Q e Qk F + ku e U k F q, ki, D, k F + ki, D, k F +èkd, Ik F + kd, Ik F è + ki, D, Similarly, wehave k F : è! kq, QkF e = kè Q; e U e è èq, QèkF e eq = Q, I eu Q F q, ki, D, k F + ki, D, F k +èkd, Ik F + kd, Ik F è + kd, Ik F : Theorem below follows from è8è and è9è. Theorem Let A and A e be two m n èm énè complex matrices having full column rank and with the polar decompositions è4è. Then kq, e QkF ki, D, k F + ki, D, k F n +èki, D k F + ki, D k F è + min ki, D, k F ; ki, D k F p 3 q ki, D k F + ki, D, k F + ki, D k + ki, D, F k F : oi è8è è9è 5
Now we are in the position to apply Theorem to perturbations for oneside scaling èfrom the leftè. Here we consider two n n nonsingular matrices G = D B and e G = D e B, where D is a scaling matrix and usually diagonal èbut this is not necessary to the theorem that followsè. B is nonsingular and usually better conditioned than G itself. Set B def = B e, B: eb is also nonsingular by the condition kbk kb, k assumed henceforth. Notice that é which will be eg = D e B = D èb +Bè =D BèI + B, èbèè = GèI + B, èbèè: So applying Theorem with D = 0 and D = I + B, èbè leads to Theorem 3 Let G = D B and e G = D e B be two nn nonsingular matrices, and let G = QH and e G = e Q e H be their polar decompositions. Set B def = B e, B. IfkBk kb, k é then kq, e QkF r kb, èbèk F + I, èi + B, èbèè, F s + è,kb, k kbk è kb, k kbk F : One can deal with one-side scaling from the right in the similar way. Acknowledgement: I thank Professor W. Kahan for his supervision and Professor J. Demmel for valuable discussions. References ëë A. Barrlund. Perturbation bounds on the polar decomposition. BIT, 30:0í3, 990. ëë C.-H. Chen and J.-G. Sun. Perturbation bounds for the polar factors. J. Comp. Math., 7:397í40, 989. ë3ë N. J. Higham. Computing the polar decompositioníwith applications. SIAM Journal on Scientic and Statistical Computing, 7:60í74, 986. 6
ë4ë C. Kenney and A. J. Laub. Polar decompostion and matrix sign function condition estimates. SIAM Journal on Scientic and Statistical Computing, :488í504, 99. ë5ë R.-C. Li. New perturbation bounds for the unitary polar factor. Manuscript submitted to SIAM J. Matrix Anal. Appl., 993. ë6ë R.-C. Li. A perturbation bound for the generalized polar decomposition. BIT, 33:304í308, 993. ë7ë J.-Q. Mao. The perturbation analysis of the product of singular vector matrices uv h. J. Comp. Math., 4:45í48, 986. ë8ë R. Mathias. Perturbation bounds for the polar decomposition. SIAM J. Matrix Anal. Appl., 4:588í597, 993. ë9ë J.-G. Sun and C.-H. Chen. Generalized polar decomposition. Math. Numer. Sinica, :6í73, 989. In Chinese. 7