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1 Relative Perturbation Bounds for the Unitary Polar actor Ren-Cang Li Mathematical Science Section Oak Ridge National Laboratory P.O. Box 2008, Bldg 602 Oak Ridge, TN LAPACK working notes è 83 èrst published July, 994, revised January, 996è Abstract Let B be an m n èm ç 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. Existing perturbation bounds for complex matrices suggest that in the worst case, the change in Q be proportional to the reciprocal of the smallest singular value of B. However, there are situations where this unitary polar factor is much more accurately determined by the data than the existing perturbation bounds would indicate. In this paper the following question is addressed: how much mayqchange if B is perturbed to e B = D BD 2? Here D and D 2 are nonsingular and close to the identity matrices of suitable dimensions. It will be proved that for such kinds of perturbations, the change in Q is bounded only by the distances from D and D 2 to identity matrices, and thus independent of the singular values of B. This material is based in part upon work supported, during January, 992íAugust, 995, by Argonne National Laboratory under grant No and by the University of Tennessee through the Advanced Research Projects Agency under contract No. DAAL03-9-C-0047, by the National Science oundation under grant No. ASC , and by the National Science Infrastructure grants No. CDA and CDA-94056, and supported, since August, 995, by a Householder ellowship in Scientic Computing at Oak Ridge National Laboratory, supported by the Applied Mathematical Sciences Research Program, Oce of Energy Research, United States Department of Energy contract DE-AC05-84OR2400 with Lockheed Martin Energy System, Inc. Part of this work was done during summer of 994 while the author was at Department of Mathematics, University of California at Berkeley.

2 Ren-Cang Li: Relative Perturbation Bounds for the Unitary Polar actor 2 Let B be an m n èm ç 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 =2 ; Q=BèB Bè,=2 ; è2è 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 2 è and V are unitary, U is mn,= and 0 = diag èç ;:::;ç n è is nonnegative. There are several published bounds stating how much the two factor matrices Q and H may change if entries of B are perturbed in arbitrary manner ë2, 3,6,8,9,, 2, 3, 4ë. In these papers, no assumption was made on how B was perturbed to B e except possibly an assumption on the smallness of kb e, Bk for some matrix norm kk. Roughly speaking, bounds in these published papers suggest that in the worst case, the change in Q be proportional to the reciprocal of the smallest singular value of B. In this paper, on the other hand, we study the change in Q, assuming B is complex and perturbed to B e = D BD 2, where D and D 2 are nonsingular and close to the identity matrices of suitable dimensions. Such perturbations covers, for example, component-wise relative perturbations to entries of symmetric tridiagonal matrices with zero diagonal ë5, 7ë, entries of bidiagonal and biacyclic matrices ë, 4, 5ë. Our results indicate that under such perturbations, the change in Q is independent of the smallest singular value of B. Assume that B has full column rank and so does B e = D BD 2. Let be the polar decompositions of B and e B respectively, and let B = QH; e B = e Q e H è4è B = UV ; e B = e U e e V è5è be the SVDs of B and e B, respectively, where e U =è e U ; e U 2 è, e U is m n, and e = and e = diag èeç ;:::;eç n è. Assume as usual that è e 0 ç ç ç ç n é0; and eç ç ç eç n é0: è6è It follows from è2è and è5è that Q = U V and e Q = e U e V.!

3 Ren-Cang Li: Relative Perturbation Bounds for the Unitary Polar actor 3 In what follows, kxk denotes the robenius norm which is the square root of the trace of X X. Notice that and eu è e B, BèV = e e V V, e U U; eu è e B, BèV = e U èd BD 2, D B + D B, BèV = e U h ebèi, D, 2 è+èd,ièb iv = e e V èi,d, 2 èv + e U èd,ièu; U è e B, Bè e V = U e U e, V e V; U èb,bè e V e = U èd BD 2, BD 2 + BD 2, BèV e h = U èi, D, è e B + BèD 2, Iè i V e to obtain two perturbation equations: = U èi, D, è e U e +V èd 2,Iè e V: e V e V, U e U = e V e èi, D, 2 èv + U e èd, IèU; è7è U U e e, V V e = U èi, D, U+V e èd 2,IèV: e è8è The rst n rows of equation è7è yields e e V V, e U U = e e V èi, D, 2 èv + e U èd, IèU : è9è The rst n rows of equation è8è yields U e U e, V V e = U èi, D, è e U e + V èd 2,IèV; e on taking conjugate transpose of which, one has e e U U, e V V = e e U èi, D, èu + e V èd 2, IèV : è0è Now subtracting è0è from è9è leads to e è e U U, e V V è+è e U U, e V Vè = e h eu èi, D, èu, e V èi, D, 2 èv i + To continue, we need a lemma from Li ë0ë. h ev èd 2, IèV, U e i èd, IèU : Lemma Let 2 C ss and, 2 C tt be two Hermitian matrices, and let E; 2 C st. If çèè T çè,è = ;, where çè è is the spectrum of a matrix, then q matrix equation X, X, =E+,has a unique solution, and moreover kxk ç kek 2 +. k k2 ç, where ç def = min!2çèè;2çè,è j!,j pj!j 2 +jj 2. èè

4 Ren-Cang Li: Relative Perturbation Bounds for the Unitary Polar actor 4 Apply this lemma to equation èè with = e,,=,, and X = e U U, e V V = e V è e V e U U V, IèV = e V è e Q Q, IèV; min E = U e èi, D, èu, V e èi, D, 2 èv; ee = V e èd 2, IèV, U e èd, IèU q kek 2 + k Ek e 2 =ç, where ç = to get kxk = k e Q Q, Ik ç q eç i +ç j çi; jçn eç 2 i +ç2 j Theorem Let B and B e = D BD 2 be two m n èm ç nè complex matrices having full column rank and with polar decompositions è4è. Then r ç kq, Qk e 2 ç ki, D, k + ki, D, 2 ç k +èkd2,ik +kd,ik è 2 ; è2è p q ç 2 ki, D, k2 + ki, D, 2 k2 + kd 2, Ik 2 + kd, Ik 2 : è3è Proof: Inequality è3è follows from è2è by the fact that è+è 2 ç 2è è for ; ç 0. Now we prove è2è. When m = n, both Q and e Q are unitary. Thus k e Q Q, Ik = k e Q èq, e Qèk = kq, e Qk. Inequality è2è is a consequence of kek çki,d, k +ki,d, 2 k and k e Ek çkd 2,Ik +kd,ik : è4è or the case mén, the last m, n rows of equation è8è produce that U 2 e U e =U 2èI,D, è e U e : Since e B has full column rank, e is nonsingular. We haveu 2 e U =U 2 èi,d, è e U : Thus ç. ku 2 e U k çku 2 èi,d, èk = kèi, D, èu 2k : è5è Notice that èu V ;U 2 è=èq; U 2 è is unitary. Hence U 2 Q = 0, and è kq, Qk e = kèq; U 2 è èq, Qèk e I, Q = e! Q,U e 2 Q = qki, Q Qk e 2 + e q k,u2 U V e k 2 ç kek 2 +ke Ek 2 e +ku 2U k 2: è6è By è5è, we get kek 2 + ku e ç ç 2 2 U k 2 ç kèi, D, èu k + ki, D, 2 k + kèi, D, èu 2k 2 = kèi, D, èu k 2 +2kèI,D, èu k ki,d, 2 k +ki,d, 2 k2 +kèi,d, èu 2k 2 ç kèi,d, èu k 2 +kèi,d, èu 2k 2 +2kI, D, k ki, D, 2 k + ki, D, 2 k2 = ki, D, k2 +2kI,D, k ki,d, 2 k +ki,d, 2 k2 = ç 2: ki,d, k +ki,d, 2 ç k

5 Ren-Cang Li: Relative Perturbation Bounds for the Unitary Polar actor 5 Inequality è2è is a consequence of è6è, è4è, and the above inequalities. Now we are in the position to apply Theorem to perturbations for one-side scaling. Here we consider two n n nonsingular matrices B = S G and B e = S G, e where S is a scaling matrix and usually diagonal. èbut this is not necessary to the theorem below.è The elements of S can vary wildly. G is nonsingular and usually better conditioned than B itself. Set G def = G e, G: eg is guaranteed nonsingular if kgk 2 kg, k 2 é which will be assumed henceforth. Notice that eb = S e G = S èg +Gè=S GèI+G, ègèè = BèI + G, ègèè: So applying Theorem with D = 0 and D 2 = I + G, ègè leads to Theorem 2 Let B = S G and e B = S e G be two n n nonsingular matrices with the polar decompositions è4è. If kgk 2 kg, k 2 é then kq, e Qk ç ç r kg, ègèk 2 + I, èi + G, ègèè, 2 ; s + è,kg, k 2 kgk 2 è 2 kg, k 2 kgk : One can deal with one-side scaling from the right in the same way. Acknowledgement: I thank Professor W. Kahan for his supervision and Professor J. Demmel for valuable discussions.

6 Ren-Cang Li: Relative Perturbation Bounds for the Unitary Polar actor 6 References ëë J. Barlow and J. Demmel. Computing accurate eigensystems of scaled diagonally dominant matrices. SIAM Journal on Numerical Analysis, 27:762í79, 990. ë2ë A. Barrlund. Perturbation bounds on the polar decomposition BIT, 30:0í3, ë3ë C.-H. Chen and J.-G. Sun. Math., 7:397í40, 989. Perturbation bounds for the polar factors. J. Comp. ë4ë J. Demmel and W. Gragg. On computing accurate singular values and eigenvalues of matrices with acyclic graphs. Linear Algebra and its Application, 85:203í27, 993. ë5ë J. Demmel and W. Kahan. Accurate singular values of bidiagonal matrices. SIAM Journal on Scientic and Statistical Computing, :873í92, 990. ë6ë N. J. Higham. Computing the polar decompositioníwith applications. SIAM Journal on Scientic and Statistical Computing, 7:60í74, 986. ë7ë W. Kahan. Accurate eigenvalues of a symmetric tridiagonal matrix. Technical Report CS4, Computer Science Department, Stanford University, Stanford, CA, 966. èrevised June 968è. ë8ë C. Kenney and A. J. Laub. Polar decompostion and matrix sign function condition estimates. SIAM Journal on Scientic and Statistical Computing, 2:488í504, 99. ë9ë R.-C. Li. A perturbation bound for the generalized polar decomposition. BIT, 33:304í 308, 993. ë0ë R.-C. Li. Relative perturbation theory: èiiè eigenspace and singular subspace variations. Technical Report UCBèèCSD , Computer Science Division, Department of EECS, University of California at Berkeley, 994. èrevised January 996è. ëë R.-C. Li. New perturbation bounds for the unitary polar factor. SIAM J. Matrix Anal. Appl., 6:327í332, 995. ë2ë J.-Q. Mao. The perturbation analysis of the product of singular vector matrices UV H. J. Comp. Math., 4:245í248, 986. ë3ë R. Mathias. Perturbation bounds for the polar decomposition. SIAM J. Matrix Anal. Appl., 4:588í597, 993. ë4ë J.-G. Sun and C.-H. Chen. Generalized polar decomposition. Math. Numer. Sinica, :262í273, 989. In Chinese.

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