A note on the Frobenius conditional number with positive definite matrices

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1 Li et al. Joural of Iequalities ad Applicatios 011, 011:10 RESEARCH Ope Access A ote o the Frobeius coditioal umber with positive defiite matrices Hogyi Li, Zogsheg Gao ad Di Zhao * * Correspodece: zdhyl010@163. com LMIB, School of Mathematics ad System Sciece, Beihag Uiversity, Beijig , P.R. Chia Abstract I this article, we focus o the lower bouds of the Frobeius coditio umber. Usig the geeralized Schwarz iequality, we preset some lower bouds for the Frobeius coditio umber of a positive defiite matrix depedig o its trace, determiat, ad Frobeius orm. Also, we give some results o a kid of matrices with special structure, the positive defiite matrices with cetrosymmetric structure. 1 Itroductio ad prelimiaries I this article, we use the followig otatios. Let C ad R be the space of complex ad real matrices, respectively. The idetity matrix i C is deoted by I = I.LetA T, Ā, A H, ad tr(a deote the traspose, the cojugate, the cojugate traspose, ad the trace of a matrix A, respectively. Re(a stads for the real part of a umber a. The Frobeius ier product <, > F i C m is defied as <A,B > F =Re(tr (B H A, for A,B Î C m,<a,b > is the real part of the trace of B H A. The iduced matrix orm is A F = < A, A> F = Re (tr(a H A = tr(a H A, which is called the Frobeius (Euclidea orm. The Frobeius ier product allows us to defie the cosie of the agle betwee two give real matrices as cos(a, B = < A, B> F A F B F. (1:1 The cosie of the agle betwee two real depeds o the Frobeius ier product ad the Frobeius orms of give matrices. A matrix A Î C is Hermitia if A H = A. A Hermitia matrix A is said to be positive semidefiite or oegative defiite, writte as A 0, if (see, e.g., [[1], p. 159] x H Ax 0, x C, (1: A is further called positive defiite, symbolized A > 0, if the strict iequality i (1. holds for all ozero x Î C. A equivalet coditio for A Î C to be positive defiite is that A is Hermitia ad all eigevalues of A are positive real umbers. The quaity { A A μ(a = 1, ifa is osigular; if A is sigular. (1:3 011 Li et al; licesee Spriger. This is a Ope Access article distributed uder the terms of the Creative Commos Attributio Licese ( which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited.

2 Li et al. Joural of Iequalities ad Applicatios 011, 011:10 Page of 9 is called the coditio umber of matrix μ(a with respect to the matrix orm. Notice that μ(a = A -1 A A -1 A = I 1 for ay matrix orm (see, e.g., [[], p. 336]. The coditio umber μ(a of a osigular matrix A plays a importat role i the umerical solutio of liear systems sice it measures the sesitivity of the solutio of liear system Ax = b to the perturbatios o A ad b. There are several methods that allow to fid good approximatios of the coditio umber of a geeral square matrix. We first itroduce some iequalities. Buzao [3] obtaied the followig extesio of the celebrated Schwarz iequality i a real or complex ier product space (H, <, >. Lemma 1.1 ([3]. For ay a, b, x Î H, there is < a, x >< x, b > 1 ( a b + < a.b > x. (1:4 It is clear that for a = b, (1.4 becomes the stadard Schwarz iequality < a, x> a x, a, x H (1:5 with equality if ad oly if there exists a scalar l such that x = la. Also Dragomir [4] has stated the followig iequality. Lemma 1. [4]. For ay a,b,x Î H, ad x 0, there is the followig < a, x >< x, b > x < a, b > a b (1:6 Daa [5] showed the followig iequality by usig arithmetic-geometric iequality. Lemma 1.3 [5]. For -square positive defiite matrices A ad B, (deta det B m/ tr(a m B m, (1:7 where m is a positive iteger. By takig A = I, B = A -1, ad m = 1 i (1.7, we obtai (deti det A 1 1/ tr(i A 1, (1:8 ( 1 1/ tr(a 1 (1:9 deta I [6], Türkme ad Ulukök proposed the followig, Lemma 1.4 [6]. Let both A ad B be -square positive defiite matrices, the cos(a, Icos(B, I 1 cos(a, B+1], (1:10 cos(a, A 1 cos(a, Icos(A 1, I 1 [cos(a, A 1 +1] 1, (1:11 cos(a, I 1, cos(a 1, I 1, (1:1 As a cosequece, i the followig sectio, we give some bouds for the Frobeius coditio umbers by cosiderig iequalities give i this sectio.

3 Li et al. Joural of Iequalities ad Applicatios 011, 011:10 Page 3 of 9 Mai results Theorem.1. Let A be a positive defiite real matrix, a be ay real umber. The, tra 1+α tra α (deta (1 α/ μ F(A; (:1 tra1+α tra α 1 tra α μ F (A. (: where μ F (A is the Frobeius coditioal umber of A. Proof. LetX, A, B be positive defiite real matrices. From Lemma 1., we have the followig < A, X> F < X, B> F X F < A, B> F A F B F, (:3 tr(a T Xtr(X T B X F tr(at B A F B F. (:4 Let B = A -1, the (.4 turs ito tr(a T Xtr(X T A 1 X tr(at A 1 F A F A 1 F. (:5 Sice both X ad A are positive defiite, we have tr(axtr(xa 1 X F A F A 1 F = μ F(A, (:6 where μ F (A is the Frobeius coditio umber of A. By takig X = A a (a is a arbitrary real umber ito (.6, there exists tra 1+α tra (1 α tra α μ F(A. (:7 Thus, it follows that tra 1+α tra (1 α tra α μ F(A, (:8 tra1+α tra α 1 tra α μ F (A. (:9 From (1.9, by replacig A with A 1-a, we get 1 0 < (det A (1 α/ tr(a (1 α. (:10

4 Li et al. Joural of Iequalities ad Applicatios 011, 011:10 Page 4 of 9 Takig (.10 ito (.8, we ca write tra1+α tra α 1 (det A (1 α/ μ F(A, (:11 tra 1+α tra α (deta (1 α/ μ F(A. (:1 I particular, let a = 1, ad by takig it ito (.7, we have the followig tra tri tra μ F(A, (:13 μ F (A. (:14 Note, whe α = 1, (.1 becomes tra 3/ tra (det A 1/ μ F(A, (:15 Takig a = 1 ito (.1, we obtai that tra (det A 1/ μ F(A. (:16 (.15, (.16 ca be foud i [6]. Example.. [ ] 0 A = 0 1. Here tra =.5,detA =1ad =. The, from (.15 ad (.16, we obtai two lower bouds of μ F (A: μ F (A tra (deta 1/ = , ad μ tra F(A (det A tra 3/ 1/ =3. Takig a = 1/4 ito (.1 ad (. from Theorem.1, aother two lower bouds are obtaied as follows: tra 1+α μ F (A tra α (det A (1 α/ = , ad μ F(A tra1+α tra α 1 tra α = I fact, μ F (A = 4.5. Thus, Theorem.1 is ideed a geeralizatio of (.15 ad (.16 give i [6]. Lemma.3. Let a 1, a,..., a be positive umbers, ad f (x =a x 1 + ax + + ax + a x a x. The, f(x is mootoously icreasig for x Î [0,+.

5 Li et al. Joural of Iequalities ad Applicatios 011, 011:10 Page 5 of 9 Proof. It is obvious that f (x =a x 1 l a 1 + a x l a + + a x l a a x 1 l a 1 a x l a =la 1 (a x 1 a x 1 +la (a x a x 1 +la (a x a x. Whe a i 1, ad x Î [0, +, l a i 0, a x i 1, ad a x i 1. Thus, l a i (a x i a x i 0. Whe 0 <a i < 1, ad x Î [0, +, we have l a i < 0, 0 < a x i 1, ad a x i l a i (a x i a x i 0. Therefore, f (x 0, ad f(x is icreasig for x Î [0,. Theorem.4. Let A be a positive defiite real matrix. The 1. Thus, A 1/ F tra 1/ μ F (A. (:17 Proof. Sice A is positive defiite, from Lemma 1.4, we have That is cos(a 1/, A 1/ cos(a 1/, I. (:18 μ F (A 1/ = tr(a 1/ A 1/ tra 1/ A 1/ F (A 1/ 1. F A 1/ (:19 F Let all the positive real eigevalues of A be l 1, l,..., l > 0. The, the eigevalues of A -1 are 1/l 1,1/l,..., 1/l > 0, the eigevalues of A are λ 1, λ,..., λ > 0, adthe eigevalue of A - are λ > 0. Next, we will prove the followig by iductio. ( i=1 λ i 1, λ,..., λ ( ( ( 1/λ i λ i 1/λ i. (:0 i=1 I case = 1, it is obvious that (.0 holds. I case =, ( ( i=1 λ i i=1 1/λ i =+(λ1 /λ +(λ /λ 1. From Lemma.3, + (l 1 /l +(l /l 1 +l 1 /l + l /l 1 =(l 1 + l (1/l 1 +1/l. Thus, (.0 holds. Suppose that (.0 holds, whe = k, ( k ( k ( k ( k λ i 1/λ i λ i 1/λ i. (:1 I case = k +1, ( k+1 i=1 λ i ( k+1 i=1 1/λ i ( k = i=1 λ i + λ k+1 ( k = i=1 λ i ( k ( k i=1 1/λ i i=1 1/λ i +1/λ k+1 + k (λ i/λ k+1 + k (λ k+1/λ i +1. (:

6 Li et al. Joural of Iequalities ad Applicatios 011, 011:10 Page 6 of 9 By Lemma.3, we get ( k+1 ( k+1 λ i ( k ( k 1/λ i λ i 1/λ i + ( k+1 ( k+1 = λ i 1/λ i. k k (λ i /λ k+1 + (λ k+1 /λ i +1 i=1 i=1 (:3 Thus, whe = k + 1, (.0 holds. O the other had, A F = tr(a = i=1 λ i, A 1 F = tra = i=1 λ i, A 1/ F = tra = ad i=1 λ i A 1/ F = tra 1 = i=1 λ 1. Therefore, i A F A 1 F A1/ F A 1/ F, (35 μ F (A μ F (A 1/. (:5 Takig (.5 ito (.19, we obtai μ F (A μ F (A 1/ tra 1/, (:6 A 1/ F A 1/ F tra 1/ μ F (A. (:7 Remark.5. (.17 ca be exteded to ay 0 <a 1, A α F tra α μ F (A. 3 The Frobeius coditio umber of a cetrosymmetric positive defiite matrix Defiitio 3.1 (see [7]. Let A =(a ij pxq Î R p q. A is a cetrosymmetric matrix, if a ij = a p i+1,q j+1,1 i p,1 j q, or J p AJ q = A, where J =(e, e -1,..., e 1, e i deotes the uit vector with the i-th etry 1. Usig the partitio of matrix, the cetral symmetric character of a square cetrosymmetric matrix ca be described as follows (see [7]: Lemma 3.. Let A =(a ij ( =m be cetrosymmetric. The, A has the form, [ BJm CJ A = m CJ m BJ m ], P T AP = [ B Jm C 0 0 B + J m C ], (3:1 Where B,C Î C m m, P = 1 [ ] Im I m. J m J m Lemma 3.3. Let Abea ( =m cetrosymmetric positive defiite matrix with the followig form

7 Li et al. Joural of Iequalities ad Applicatios 011, 011:10 Page 7 of 9 [ ] BJm CJ A = m, CJ m BJ m where B,C Î R m m. The there exists a orthogoal matrix P such that [ ] P T M AP =, N where M = B - J m C, N = B + J m C ad M -1, N -1 are positive defiite matrices. Proof. From Lemma 3., there exists a orthogoal matrix P such that [ ] H = P T M AP =. N Sice A is positive defiite, the ay eigevalue of A is positive real umber. Thus, the eigevalues of H are all positive real umbers. That is to say, all eigevalues of M ad N are positive real umbers. Thus, M ad N arebothpositivedefiite.itis obvious that M -1 ad N -1 are positive defiite matrices. Lemma 3.4. Let A, B are positive defiite real matrices. The, tra (detb 1/ A F B 1 F. (3: Proof. Let X be positive defiite. By Lemma 1., we have < A, X> F < X, B> F X < A, B> F F A FB F. (3:3 Thus, tr(a T Xtr(X T B X F tr(at B A FB F. (3:4 By replacig X, B with I ad B -1, respectively, i iequality (3.4, we ca obtai tr(atr(b 1 tr(at B 1 A F B 1 F. (3:5 That is, tr(atr(b 1 tr(at B 1 From Schwarz iequality (1.5, A F B 1 F. (3:6 tr(a T B 1 =< A, B 1 > F A F B 1 F. (3:7 By takig (3.7 ito (3.6, we have tr(atr(b 1 A F B 1 F A F B 1 F + tr(at B 1. (3:8 From (1.9, (1/detB 1/ trb 1. (3:9

8 Li et al. Joural of Iequalities ad Applicatios 011, 011:10 Page 8 of 9 Therefore, tra (det B 1/ A F B 1 F. (3:10 Theorem 3.5. Let A be a cetrosymmetric positive defiite matrix with the form [ ] BJm CJ A = m, B, C R m m. CJ m BJ m Let M = B - J m C, N = B + J m C. The, ( ( ( ( μ F (A trm (det M 1/m m trn + (det N 1/m m trm trn + (det N 1/m + (det M 1/m Proof. By Lemma 3., there exists a orthogoal matrix P such that [ ] [ ] H = P T B Jm C M AP = =. B + J m C N Thus,. (3:11 μ F (A = A F A 1 F = μ F (H = H F H 1 F = M F + N F M 1 F + N 1 F. (3:1 Therefore, μ F (A =( M F + N F ( M 1 F + N 1 F From Lemma 3.4, = M F M 1 F + N F N 1 F + M F N 1 F + N F M 1 F = μ F (M+μ F (N+( M F N 1 F +( N F M 1 F. trm (det N 1/m M F N 1 F, trn (det M 1/m N F M 1 F. (3:14 From (.18, Thus, trm (det M 1/m m μ trn F(M, ad (det N 1/m m μ F(N. (3:15 ( ( ( ( μ F (A trm (det M 1/m m trn + (det N 1/m m trm trn + (det N 1/m + (det M 1/m. Example [ ] A = 0530 M 0350,adPT AP = = N 3005 [ ] 0 0 [ ]. Here =4,m =, det (A = 56, tr A = 0, tr M =4,trN = 16, det(m = 4, det(n = 64. From Theorem 3.5, a lower boud of μ F (A is as follows:

9 Li et al. Joural of Iequalities ad Applicatios 011, 011:10 Page 9 of 9 I fact ( ( ( ( μ F (A trm (detm 1/m m trn + (detn 1/m m trm trn + (detn 1/m + (detm 1/m =8.5. μ F (A = μ F (M+μ F (N+( M F N 1 F +( N F M 1 F =8.5. O the other had, the lower bouds of μ F (A i (.15 ad (.16 provided by [6] are μ F (A tra (deta 1/ = , ad μ tra F(A =6. 1/ (det A tra 3/ It ca easily be see that, i this example, the best lower boud is the first oe give by Theorem 3.5. Ackowledgemets The authors would like to thak two aoymous referees for readig this article carefully, providig valuable suggestios ad commets which have bee implemeted i this revised versio. This study was supported by the Natioal Sciece Foudatio of Chia No ad the Natio Sciece Foudatio of Chia No Authors cotributios Hogyi Li carried out studies o the liear algebra ad matrix theory with applicatios, ad drafted the mauscript. Zogsheg Gao read the mauscript carefully ad gave valuable suggestios ad commets. Di Zhao provided umerical examples, which demostrated the mai results of this article. All authors read ad approved the fial mauscript. Competig iterests The authors declare that they have o competig iterests. Received: 1 Jue 011 Accepted: 4 November 011 Published: 4 November 011 Refereces 1. Zhag, F: Matrix Theory: Basic Results ad Techiques. Spriger, New York (1999. Hor, RA, Johso, CR: Matrix Aalysis. Cambridge Uiversity Press, Cambridge ( Buzao, ML: Gemeralizzazioe della diseguagliaza di Cauchy-Schwarz. Redicoti del Semiario Matematico Uiversita e Politecico di Torio 31, (1971. (1974 (Italia 4. Dragomir, SS: Refiemes of Buzao s ad Kurepa s iequalities i ier product spaces. Facta Uiversitatis. 0, ( Daa, FM: Matrix ad operator iequalites. J Iequal Pure Appl Math (3 (001. article Ramaza, T, Zübeyde, U: O the Frobeius coditio umber of positive defiite matrices. J Iequal Appl (010. Article ID ( Liu, ZY: Some properties of cetrosymmetric matrices. Appl Math Comput. 141, 17 6 (00 doi: /109-4x Cite this article as: Li et al.: A ote o the Frobeius coditioal umber with positive defiite matrices. Joural of Iequalities ad Applicatios :10. Submit your mauscript to a joural ad beefit from: 7 Coveiet olie submissio 7 Rigorous peer review 7 Immediate publicatio o acceptace 7 Ope access: articles freely available olie 7 High visibility withi the field 7 Retaiig the copyright to your article Submit your ext mauscript at 7 sprigerope.com