Path product and inverse M-matrices

Transcription

1 Electronic Journl of Liner Algebr Volume 22 Volume 22 (2011) Article Pth product nd inverse M-mtrices Yn Zhu Cheng-Yi Zhng Jun Liu Follow this nd dditionl works t: Recommended Cittion Zhu, Yn; Zhng, Cheng-Yi; nd Liu, Jun. (2011), "Pth product nd inverse M-mtrices", Electronic Journl of Liner Algebr, Volume 22. DOI: This Article is brought to you for free nd open ccess by Wyoming Scholrs Repository. It hs been ccepted for inclusion in Electronic Journl of Liner Algebr by n uthorized editor of Wyoming Scholrs Repository. For more informtion, plese contct scholcom@uwyo.edu.

2 PATH PRODUCT AND INVERSE M-MATRICES YAN ZHU, CHENG-YI ZHANG, AND JUN LIU Abstrct. It is known tht inverse M-mtrices re strict pth product (SPP) mtrices, nd tht the converse is not true for mtrices of order greter thn 3. In this pper, given normlized SPP-mtrix A, some new vlues s for which A+s I is n inverse M-mtrix re obtined. Our vlues s extend the vlues s given by Johnson nd Smith C.R. Johnson nd R.L. Smith. Positive, pth product, nd inverse M-mtrices. Liner Algebr Appl., 421: , The question whether or not 4 4 SPP-mtrix is P-mtrix is settled. Key words. M-mtrix, Inverse M-mtrix, Pth product mtrix, P-mtrix. AMS subject clssifictions. 15A48, 15A Introduction. An n n mtrix A = ( ij ) is n M-mtrix if ij 0 (i j) nd A 1 0. A nonnegtive mtrix which is the inverse of n M-mtrix is n inverse M-mtrix (IM-mtrix). Inverse M-mtrices rise in mthemticl modeling, rndom energy models in sttisticl physics 1, numericl integrtion nd the Ising model of ferromgnetism 12. There hs been gret del of work on specil types of IM-mtrices (see, for exmple, 3, 4, 9 11). Here we will be interested in the property ij jk (1.1) ik, 1 i,j,k n jj of n IM-mtrix A = ( ij ) n n, n 3, which ws first noted in 12 nd more fully developed in 7. Following 7, we cll (1.1) the pth product conditions or PP conditions, for short. An n n nonnegtive mtrix A = ( ij ), with ii > 0, stisfying these conditions is Received by the editors on September 6, Accepted for publiction on June 27, Hndling Editor: Joo Felipe Queiro. College of Mthemtic nd Informtion Science, Qujing Norml University, Qujing, Yunnn, , P.R. Chin (zhuynlj@163.com). Supported by the Science foundtion of Qujing Norml University (No. 2009QN017). Deprtment of Mthemtics nd Mechnics of School of Science, Xi n Polytechnic University, Xi n, Shnxi , P.R. Chin. Supported by the Science Foundtion of the Eduction Deprtment of Shnxi Province of Chin (No. 11JK0492) nd the Scientific Reserch Foundtion of Xi n Polytechnic University (No. BS1014). College of Mthemtic nd Informtion Science, Qujing Norml University, Qujing, Yunnn, , P.R. Chin. Supported by the Ntionl Nturl Science Foundtion of Chin (No ) nd Yunnn NSF Grnt (No. 2010CD086). 644

3 Pth Product nd Inverse M-Mtrices 645 PP-mtrix. Moreover, if t lest one strict inequlity in (1.1) holds for i = k nd i j, then A is strict pth product (SPP) mtrix. In 7 (see lso 12), it is proved tht n IM-mtrix is n SPP-mtrix. Furthermore, n SPP-mtrix is n IM-mtrix when n 3, nd this is not necessrily the cse for lrger n. Consequently, it ws noted in 6 tht n SPP-mtrix my be mde n IM-mtrix by dding n pproprite nonnegtive digonl mtrix. We sy tht n n n nonnegtive mtrix A = ( ij ) is normlized if ii = 1 nd ij < 1, for i j. It ws noted in 7 tht if A is n n n SPP-mtrix, then there exist positive digonl mtrices D nd E such tht B = DAE, where B is normlized SPP-mtrix. Given n n n mtrix A nd index sets α, β N, N = {1,...,n}, we denote by Aα,β the submtrix lying in rows α nd columns β. Similrly, A(α,β) denotes the submtrix deleting rows α nd columns β. If α = β, then we denote the principl submtrix Aα,α (resp., A(α,α)) by Aα (resp., A(α)). An lmost principl submtrix (resp., minor) is submtrix Aα, β (resp., Aα,β) for which α nd β hve the sme number of elements nd differ just in one of their elements. Almost principl minors re exctly the numertors of offdigonl entries of inverses of principl submtrices. Following 8, we bbrevite lmost principl minor to APM. In this pper, for n n n normlized SPP-mtrix A = ( ij ), we will give new vlues s such tht A + s I is n IM-mtrix. Our vlues s extend the vlues given by Johnson nd Smith 6. Exmples re lso given, nd we will show tht 4 4 normlized SPP-mtrix is necessrily P-mtrix; this nswers question rised in Min results. The results bout SPP-mtrices estblished by Johnson nd Smith 7 tht we shll use re the following. Lemm 2.1. Let A = ( ij ) be normlized SPP-mtrix of order n. Then Aα is normlized SPP-mtrix. Lemm 2.2. Let A = ( ij ) be normlized SPP-mtrix of order n. Then ll 3 3 principl submtrices of A re IM-mtrices. The following pper in 6. Theorem 2.3. Let A = ( ij ) be normlized SPP mtrix of order n, n 2, whose proper principl minors re positive nd whose APMs re signed s those of n IM-mtrix. Then, 1. For ech nonempty proper subset α of N = {1,2,...,n} nd for ll indices

4 646 Yn Zhu, Cheng-Yi Zhng, nd Jun Liu i α nd j / α, we hve 2. A > 0; 3. A is n IM-mtrix. Aα > mx{ Aα i + j,α, Aα,α i + j }; Theorem 2.4. Let A = ( ij ) be 4 4 normlized SPP-mtrix. Then A + I is n IM-mtrix. Furthermore, A + si need not be n IM-mtrix when s < 1. Now we re redy to stte the following result bout 4 4 normlized SPP mtrices. Theorem 2.5. Let A = ( ij ) be 4 4 normlized SPP-mtrix. Then A + s I is n IM-mtrix for ll s m, where ik kj m = mx 1, k = 1,...,n, k i,j, nd ij 0. i j ij Proof. Following the ide of Theorem 2.4, to show A + mi is n IM-mtrix, we will show tht the (4,1) APM (i.e., the erminnt of A{1,2,3}, {2,3,4}) is nonnegtive. Note tht (A + mi)(4,1) = m m 34 = (1 + m) 2 14 (1 + m) (1 + m) = (1 + m)( m ) (1 + m)( m ) , where m = 14 (m ) 0. If the sum of the lst three terms is nonnegtive, then the erminnt is nonnegtive by the pth product inequlities. Otherwise, we hve (A + mi)(4,1) (1 + m)( m ) = (1 + m)( m ) +( ) (1 + m)( m ) + ( ) = m m( m )

5 Pth Product nd Inverse M-Mtrices 647 As consequence, A + mi is n IM-mtrix. Since s m, A + s I is necessrily n IM-mtrix. Exmple 2.6. Consider the following normlized SPP-mtrix A = Then A is not n IM-mtrix, since A(2, 1) = By ctul clcultion, m = = 0.875, so A I is n IM-mtrix. In fct, A + mi is n IM-mtrix if nd only if m For convenience, let n 3, nd, for i j, define 1 u ij (A) = ij ik kj, ij 0, k=1,k i,j 0, ij = 0, U(A) = mx i j u ij (A), i.e., the lrgest vlue mong u ij (A), where i j, u(a) the second lrgest vlue mong u ij (A), where i j, ε = U(A) u(a), ε = U(Aα) u(aα). In 6, Theorem 3, lower bound is given for the numbers s such tht A + si is n IM-mtrix. If U(A) > 1, then this bound is zero nd it cnnot be improved. But for U(A) 1 Theorem 2.7 improves the lower bound U(A) 1 given in 6, Theorem 3. Theorem 2.7. Let A = ( ij ) be normlized SPP mtrix of order n, n 3, nd let l = mx{u(a),1}. Then A + s I is n IM-mtrix for ll s l ε 1. Proof. We use proof technique nlogous to tht in 6, Theorem 3, nd induction on n. If n = 3, A is n IM-mtrix nd thus A + s I is n IM-mtrix for ll When n > 3, proceeding inductively, let s l ε 1. C = A + s I = (c ij ) n n. It follows tht the (n 1) (n 1) principl minors of C re positive since for ny principl submtrix Aα of A, Aα + s I is n IM-mtrix so tht Aα + s I is n IM-mtrix, s s s, where { 0, U(Aα) 1, s = U(Aα) ε 1, U(Aα) > 1.

6 648 Yn Zhu, Cheng-Yi Zhng, nd Jun Liu Using Theorem 2.3 nd permuttion similrity, it is enough to prove tht the complement of the (1,2)-entry is nonnegtive, tht is, c 21 C({1,2}) c 23 c 2n dj C({1,2}) c 31.. c n1 0, or c 21 C({1,2}) c 23 c 2n dj C({1,2}) c 31.. c n1. Dividing by C({1,2}), we obtin (2.1) c 21 c 23 c 2n C({1,2}) 1 Let b ij, i,j = 3,...,n, be the entries of C({1,2}) 1. By induction, we verify tht C 1 = B = (b ij ) is n M-mtrix. Obviously, the right hnd side of (2.1) is i,j=3 c 2i b ij c j1 = i j Since b ij 0, by pth product c 2i b ij c j1 + c 31. c n1. c 2i b ii c i1. c 2i b ij c j1 c 2i b ij c ji c i1 ; i j i j pplying Fischer s inequlity 5 to the IM-mtrix C({1,2}), we hve So C({1,2}) c ii C({1,2,i}) = (1 + s )C({1,2,i}). From the bove inequlities, we obtin j=3 c 2i b ij c j1 = j=3,j i 1 C({1,2,i}) = b 1 + s ii. C({1,2}) c 2i b ij c j1 + (c 2i b ii c i1 + c 2i b ii c ii c i1 c 2i b ii c ii c i1 ).

7 Pth Product nd Inverse M-Mtrices 649 Since c j1 = j1 ji i1 = c ji c i1 0 nd b ij 0, i j, we obtin c 2i b ij c j1 j=3 n c 2i b ij c ji c i1 + n (1 c ii )c 2i b ii c i1 j=3 = n c 2i c i1 b ij c ji + n ( s )c 2i b ii c i1. j=3 Observing tht n j=3 b ijc ji = 1, the (i,i) entry of BB 1, we get c 2i b ij c j1 j=3 n c 2i c i1 (1 + ( s )b ii ) n c 2i c i1 (1 + ( s ) 1 1+s ) = 1 1+s n c 2ic i1 = 1 1+s 2i i1 1 1+s (U(A) ε) 21 = 21 = c 21. Exmple Consider the 4 4 normlized SPP-mtrix A = As seen in 12, A is not n IM-mtrix (the (2,3)-entry of A 1 is positive). By ctul clcultion, U(A) = 1 31 ( ) = 1.7 > 1. Hence, A + si is IM for ll s 0.7 ccording to Theorem 3 of 6. However, ε = mx{0,(u(a) u(a))}= So ccording to Theorem 2.7 A+s I is n IM-mtrix for ll s (In fct, A + s I is n IM-mtrix if nd only s 0.18.). 6. Remrk 2.9. If U(A) = u(a), then Theorem 2.7 is the sme s Theorem 3 of Similr to 6, Theorem 4, we hve: Theorem Let A = ( ij ) be normlized SPP mtrix of order n, n 3. Then A + s I is n IM-mtrix for ll s n 3 ε. Proof. The result follows from Lemm 2.2 (ii) of 6 nd Theorem 2.7. A consequence of Theorem 2.10 is s follows.

8 650 Yn Zhu, Cheng-Yi Zhng, nd Jun Liu Corollry Let A = ( ij ) be n n n nonnegtive mtrix with positive digonl entries nd let D nd E be positive digonl mtrices such tht DE = n 3 ε dig(a) 1. Then, if DAE n 3 ε I is n SPP-mtrix, A is n IM-mtrix. Following 6, the Hdmrd dul of the IM-mtrices, denoted by IM D, is defined to be the set of ll mtrices B such tht A B is n IM-mtrix for ll IM-mtrices A. We my obtin the following results which re similr to those in 6. Lemm Let A = ( ij ) be normlized IM-mtrix of order n. Then A + n 3 ε I IM D. Theorem Let A = ( ij ) be n IM-mtrix of order n nd let D nd E be positive digonl mtrices such tht A 1 = DAE is normlized. Then A + n 3 ε D 1 E 1 IM D. A rel n n mtrix A is clled P-mtrix if the principl minors of A re ll positive. Obviously, IM-mtrices re P-mtrices. SPP-mtrices re not necessrily P-mtrices for n 6, but for n 3 they re 7. Here we will nswer the question whether 4 4 SPP-mtrix is P-mtrix or not. We need the following lemm 2, Lemm 2.3. Lemm Let A = ( ij ) be n IM-mtrix of order n, whose columns re denoted by α 1,α 2,...,α n. Then for ny x = (x 1,x 2,...,x n ) T, the functions f(x) = (α 1,α 2,...,α n 1,x) nd g(x) = (x,α 2,...,α n 1,α n ) hve the following properties: 1) If x = (x 1,x 2,...,x n ) T y = (y 1,y 2,...,y n ) T nd x n = y n, then it holds tht f(x) f(y); 2) If x = (x 1,x 2,...,x n ) T y = (y 1,y 2,...,y n ) T nd x 1 = y 1, then it holds tht g(x) g(y). Theorem Let A = ( ij ) be 4 4 SPP mtrix. Then A is P-mtrix. Proof. Recll tht P-mtrix is rel n n mtrix whose principl minors re ll positive. From Lemm 2.1 nd Lemm 2.2, we know tht ll 2 2 nd 3 3 principl minors of A re positive. It suffices to prove tht A > 0.

9 Pth Product nd Inverse M-Mtrices 651 Set α = {2,3} = N \ {1,4}, nd let A be prtitioned s 11 A1,α 14 A = Aα, 1 Aα Aα, A4,α 44 We hve b 14 = ( 1) 4+1 A1,α 14 Aα Aα, 4 Aα,1 Aα b 41 = ( 1) A4,α = 14 Aα, 4 A1,α Aα Aα Aα,1 = A4,α 41 If b 14 b 41 0, then from (1.5) of 8 nd Aα > 0, we hve A > 0. If b 14 b 41 0, since i i4, i i1 ( i α), we obtin 14 Aα,1 11 Aα,4, 41 Aα,4 44 Aα,1. From Lemm 2.2, we observe tht ech principl submtrix A of order 3 is n inverse M-mtrix. According to Lemm 2.14, we deduce tht A1,α = A1,α Aα, 1 Aα 14 Aα,1 Aα A1,α 11 Aα,4 Aα = A1,α. Aα, 4 Aα Similrly, Aα Aα,4 41 A4,α 44 Aα 41 Aα,4 = A4,α Aα 44 Aα,1 A4,α Aα Aα,1 = 44 A4,α 41 By the bove inequlities, we hve A1,α 14 Aα,1 Aα Aα Aα, 4 41 A4,α = ( 1) n 2 14 A1,α Aα Aα,1 ( 1) n 2 Aα, 4 Aα A4,α 41 = A1,α Aα Aα,1 44 Aα, 4 Aα A4,α A1,α Aα Aα,4. Aα, 1 Aα A4,α 44.,.

10 652 Yn Zhu, Cheng-Yi Zhng, nd Jun Liu Applying (1.5) of 8, it follows tht A Aα = 11 A1,α Aα Aα,4 Aα, 1 Aα A4,α 44 A1,α 14 Aα,1 Aα Aα Aα, 4 41 A4,α ( ) A1,α Aα Aα,4 Aα, 1 Aα A4,α 44 > 0. Consequently, A > 0, ll 2 2 nd 3 3 principl minors of A re positive, so A is P-mtrix. Acknowledgment. The uthors would like to thnk very much Professor Joo Queiro nd n nonymous referee for their iled nd helpful suggestions for revising this mnuscript. REFERENCES 1 D. Cpocci, M. Cssndro, nd P. Picco. On the existence of thermodynmics for the generlized rndom energy model. J. Sttist. Phys., 46: , S.C. Chen. A property concerning the Hdmrd powers of inverse M-mtrices. Liner Algebr Appl., 381:53 60, C. Dellcherie, S. Mrtínez, nd J.S. Mrtín. Description of the sub-mrkov kernel ssocited to generlized ultrmetric mtrices: An lgorithmic pproch. Liner Algebr Appl., 318:1 21, M. Fiedler. Specil ultrmetric mtrices nd grphs. SIAM J. Mtrix Anl. Appl., 22: , R.A. Horn nd C.R. Johnson. Topics in Mtrix Anlysis. Cmbridge University Press, New York, C.R. Johnson nd R.L. Smith. Positive, pth product, nd inverse M-mtrices. Liner Algebr Appl., 421: , C.R. Johnson nd R.L. Smith. Pth product mtrices. Liner Multiliner Algebr, 46: , C.R. Johnson nd R.L. Smith. Aimost principl minors of inverse M-mtrices. Liner Algebr Appl., 337: , I. Koltrcht nd M. Neumnn. On the inverse M-mtrix problem for rel symmetric positivedefinite Toeplitz mtrices. SIAM J. Mtrix Anl. Appl., 12: , S. Mrtínez, J.S. Mrtín, nd X.D. Zhng. A new clss of inverse M-mtrices of tree-like type. SIAM J. Mtrix Anl. Appl., 24: , S. Mrtínez, G. Michon, nd J.S. Mrtín. Inverse of ultrmetric mtrices re of Stieltjes type. SIAM J. Mtrix Anl. Appl., 15:98 106, R.A. Willoughby. The inverse M-mtrix problem. Liner Algebr Appl., 18:75 94, 1977.