Filomat 30: 06, 37 375 DOI 0.98/FIL67M Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Multiplicative Perturbation Bounds of the Group Inverse Oblique Projection Lingsheng Meng a, Bing Zheng a a School of Mathematics Statistics, Lanzhou University, Lanzhou 730000, PR China Abstract. In this paper, the multiplicative perturbation bounds of the group inverse related oblique projection under general unitarily invariant norm are presented by using the decompositions of B # A # BB # AA #.. Introduction Let C n n be the set of all n n complex matrices. For a given matrix A C n n, the symbols A, A, A, A #, IndA will st for the conjugate transpose, the spectral norm, general unitarily invariant norm, the group inverse, the index of A, respectively. I denotes the n n identity matrix. Also, for the sake of the simplicity in the later discussion, we will adopt the following notations with A C n n : P A = AA # = A # A, P A = I P A. In addition, we always assume that D D are n n nonsingular matrices throughout this paper. We recall that the group inverse A # of a matrix A C n n is the unique solution X, if exists, to the following three equations []: AXA = A, XAX = X, 3 AX = XA. A square matrix A has a group inverse if only if ranka = ranka, i.e., IndA =. When A # exists, P A = AA # is the projector which projects a vector on RA along NA, i.e., AA # = P RA,NA, where RA NA are the range null space of A, respectively. For the details of group inverse, we refer the readers to the book by Ben-Israel Greville []. The group inverse plays an important role in numerical analysis, Markov chains, etc. see [, 8, 9]. However, in most numerical applications the elements of A will seldom be known exactly, so it is necessary to know how its group inverse is perturbed when A is perturbed. There are extensive studies in this regard, e.g., [6, 9, 0], for the so-called additive perturbations, namely A is perturbed to B = A + E. Since the matrix scaling technique is often used to yield better-conditioned problems [3], the multiplicative perturbation model B = D AD with both D D are nonsingular matrices near the identities has been received much attention. Notice that B = D AD can be rewritten as B = A + E with E = I D A D AI D 00 Mathematics Subject Classification. Primary 5A09; Secondary 65F0 Keywords. Group inverse; multiplicative perturbation bound; oblique projection. Received: 0 September 04; Accepted: 9 November 04 Communicated by Dragana Cvetković Ilić Research supported by the National Natural Science Foundation of China No. 737 Email address: bzheng@lzu.edu.cn Bing Zheng
L. Meng, B. Zheng / Filomat 30: 06, 37 375 37 or E = AI D I D AD, the multiplicative perturbation is a special additive perturbation. Hence the existing additive perturbation bounds can be applied directly to give the multiplicative perturbation bounds. But in general this technique may produce unideal perturbation bounds because it overlooks the nature of the multiplicative perturbation. Various multiplicative perturbation analysis have been done to many problems, such as the polar decomposition [4], the singular value decomposition [5], the Moore- Penrose inverse [7] when A is multiplicatively perturbed. In this paper, we will study the multiplicative perturbation bounds to the group inverse the related oblique projection under unitarily invariant norm.. Multiplicative Perturbation Bounds of the Oblique Projection In this section, we study the multiplicative perturbation bounds of the oblique projection related to group inverse under general unitarily invariant norm. Let A C n n, B = D AD IndA =, then B # may not exist see Example. After a simple analysis, we can get the following theorem. Theorem.. Let A C n n B = D AD, then B # exists if only if rankad D A = ranka. Example. Let A = 0 0 0, D = I D = 0 0 B = D AD = 0 0 0. Then It is easy to see that ranka = ranka = 0 = rankb < rankb =. Hence, B # does not exist. In order to get the multiplicative perturbation bounds of the oblique projection, we fist give a decomposition of BB # AA #. Lemma.. Let A C n n, B = D AD IndA =. If rankad D A = ranka, then P B P A = P B I D P A + P B D IP A.. Proof. Since P B P A = B # B AI AA # + I BB # B AA #. B A = BI D + D IA,.3 combining..3, we immediately get.. Based on the above lemma, we can give a multiplicative perturbation bound for P B P A. But before stating this result, we mention another lemma which will be used. Lemma.3. [0] Let A C n n with IndA =. Then AA # = I AA #. Theorem.4. Let A C n n, B = D AD IndA =. If rankad D A = ranka, then P B P A P A P B I D + I D..4 If the used norm is the spectral norm, we have. P B P A P A P B I D + I D..5
L. Meng, B. Zheng / Filomat 30: 06, 37 375 373 Proof. Note that the spectral norm is a special unitarily invariant norm.5 is a direct consequence of.4. So we only prove.4. Taking the unitarily invariant norm on both sides of. using Lemma., we get the estimate P B P A P B P A I D + P A P B I D = P A P B I D + I D. The proof is completed. Corollary.5. Under the assumptions of Theorem.4, if rankad D A = ranka γ = P A I D + I D <, then we have P A + γ P B P A γ.6 i.e., P B P A P A γ I D + I D..7 Proof. From.4, we have P B P A P B P A P A P B I D + I D, γ P B P A, since γ > 0, we get the right inequality of.6. Similarly, from P A P B P B P A, we can get the left inequality of.6. It is obvious that,.7 is the direct consequence of.4.6. 3. Multiplicative Perturbation Bounds of the Group Inverse In this section, we will provide multiplicative perturbation bounds of the group inverse under general unitarily invariant norm. To this end, we need the following lemma. Lemma 3.. Let A C n n, B = D AD such that IndA = rankad D A = ranka, then we have B # = P B D A# D P B 3. = I + Θ D, D A # I + Θ D, D, 3. where Θ D, D = P B D I P BI D, Θ D, D = D I P B I D P B D denotes the inverse of the conjugate transpose of D. Proof. Let Z be the matrix on the right side of equation 3.. From B = D AD, we have BZ = BD A# D BB# = D AA# AD B # = BB # ZB = B # BD A# D B = B# D AD = B # B. Hence, BZB = B, ZBZ = Z BZ = ZB, i.e., B # = Z. To prove 3., it is only needed to prove I + Θ D, D A # = P B D A# A # I + Θ D, D = A # D P B.
L. Meng, B. Zheng / Filomat 30: 06, 37 375 374 In fact, we have I + Θ D, D A # = [I + I B # BD I B# BI D ]AA# A # = A # I B # BA # B # BI D A# = P B D A#. Similarly, we can prove A # I + Θ D, D = A # D P B. Obviously, Lemma 3. is valid both for full rank rank deficient matrices. Combining.6 3., we can get the following corollary. Corollary 3.. With the same assumptions as in Lemma 3.. max{ I D, I D } <, we have If γ = P A I D + I D < B # P A γ ΦD, D A#, where ΦD, D = I D I D. The following corollary presents an expression for B # A # that follows directly from 3., which is very important to get the perturbation bound of group inverse. Corollary 3.3. Let Θ D, D Θ D, D be the same as in Lemma 3.. With the same assumptions as in Lemma 3., we have B # A # = Θ D, D A # + A # Θ D, D + Θ D, D A # Θ D, D. 3.3 Next, we state the main result in this section. Theorem 3.4. With the same assumptions as in Lemma 3., then B # A # A # [ P B I D + I D + I D + I D + P B I D + I D I D + I D ]. 3.4 If γ = P A I D + I D <, we have B # A # A # [ P A γ I D + I D + I D + I D Proof. From Lemma., we can get + P A γ I D + I D I D + I D ]. 3.5 Θ D, D P B I D + I D 3.6 Θ D, D P B I D + I D. 3.7 Hence, 3.4 follows directly from 3.3, 3.6 3.7. Substituting.6 into 3.4, we can get 3.5 immediately.
L. Meng, B. Zheng / Filomat 30: 06, 37 375 375 4. Numerical Comparison In this section, we give an example to illustrate that the multiplicative perturbation bound is much better, in some cases, than the additive perturbation bound. To compare the multiplicative perturbation bound with the additive perturbation bound, we first mention the additive perturbation bounds of group inverse related oblique projection which were obtained in [0]. Let A, B = A + E C n n IndA = IndB =, then B # A # + max{ P A, P B } max{ A #, B# } E 4. P B P A max{ P A, P B } max{ A #, B # } E. 4. 0 0.0 Example. Let A = 0 0.0 0, D = 0 0 D = I 3. Then 0 0 B = D AD =.0 0 0.0 0 E = B A = 0.0 The estimates of the additive multiplicative perturbation bounds of group inverse associated oblique projector can be found in the following table we choose the spectral norm: Exact value Additive bound 4. Multiplicative bound 3.4 B # A # 0.0005 60 0.000 Exact value Additive bound 4. Multiplicative bound.4 P B P A 0 0.4 0.000 Obviously, the multiplicative perturbation bounds of the group inverse the oblique projection are much better than the additive perturbation bounds for the above-mentioned example.. References [] A. Ben-Israel, T.N.E. Greville, Generalized Inverses: Theory Applications, nd edition, Springer, New York, 003. [] P. Bhimasankaram, D. Carlson, The group inverse of bordered matrix its application, Technical Report, No. 57, University of California, Santa Barbara, CA, 988. [3] X.W. Chang, R.C. Li, Multipliative perturbation analysis for QR factorizations, Numer. Algebra Control Optim. 0 30 36. [4] R.C. Li, Relative perturbation bounds for the unitary polar factor, BIT 37 997 67 75. [5] R.C. Li, Relative perturbation theory: I. eigenvalue singular value variations, SIAM J. Matrix Anal. Appl. 9 998 956 98. [6] X.Z. Li, Y. Wei, An improvement on the perturbation of the group inverse oblique projection, Linear Algebra Appl. 338 00 53 66. [7] L.S. Meng, B. Zheng, The optimal perturbation bounds of the Moore-Penrose inverse under the Frobenius norm, Linear Algebra Appl. 43 00 956 963. [8] Y. Wei, A characterization representation of the Drazin inverse, SIAM J. Matrix Anal. Appl. 7 996 744 747. [9] Y. Wei, G. Wang, The perturbation theory for the Drazin inverse its applications, Linear Algebra Appl. 58 997 79 86. [0] Y. Wei, On the perturbation of the group inverse oblique projection, Appl. Math. Comp. 98 999 9 4.