Journal of Computational and Applied Mathematics


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1 Journal of Computatonal and Appled Mathematcs 37 (03) 6 35 Contents lsts avalable at ScVerse ScenceDrect Journal of Computatonal and Appled Mathematcs journal homepage: wwwelsevercom/locate/cam The nverse of banded matrces Emrah Kılıç a Pantelmon Stanca b a TOBB Economcs and Technology Unversty Mathematcs Department Sogutozu Ankara Turkey b Naval Postgraduate School Department of Appled Mathematcs Monterey CA USA a r t c l e n f o a b s t r a c t Artcle hstory: Receved 8 June 00 Receved n revsed form 8 September 0 Keywords: Trangular matrx Hessenberg matrx Inverse rbanded matrx The nverses of rbanded matrces for r = 3 have been thoroughly nvestgated as one can see from the references we provde Let B rn ( r n) be an n n matrx of entres {a j } r r j r wth the remanng unndexed entres all zeros In ths paper generalzng a method of Mallk (999) [5] we gve the LU factorzaton and the nverse of the matrx B rn (f t exsts) Our results are vald for an arbtrary square matrx (takng r = n) and so we wll gve a new approach for computng the nverse of an nvertble square matrx Our method s based on Hessenberg submatrces assocated to B rn 0 Elsever BV All rghts reserved Background There are varous methods for fndng nverses of matrces (f these exst) and we recall the Gauss Jordan method the trangular decomposton such as LUD or Cholesky factorzaton to menton only a few A very popular approach s based A A on block parttonng Let A = whose nverse (called Schur s complement) s A A A A = A A B A A A A B B A A B where B = A A A A (under the rght condtons namely that A B are nvertble) Further an teratve approach for fndng the nverse exsts but ts convergence s a functon of the matrx s condton number cond(a) = A A (n some norm say l where A = max j a j ; or l wth A = A T ) We wll not go nto detals regardng these parameters of a matrx as the consderable lterature has been dedcated to these concepts Certanly the nverse of a dagonal matrx (f t exsts) s found easly: f A = dag(a a n ) then A = u c 0 0 dag(a 0 u c 0 a n ) If U s a bdagonal say U = u 0 then U = (v j ) j where the entres 0 u n satsfy the recurrences 0 > j; v j = = j u c v j < j u Correspondng author Emal addresses: (E Kılıç) (P Stanca) /$ see front matter 0 Elsever BV All rghts reserved do:006/jcam00708
2 E Kılıç P Stanca / Journal of Computatonal and Appled Mathematcs 37 (03) These recurrences can certanly be solved and we obtan 0 > j v j = j ck j v j k= u k Gong further to trdagonal matrces thngs start to change We want to menton here the work of Schlegel [] who showed b 0 0 b 0 that f B = then the nverse B = (c j ) j s gven by c j = r r r n j and c j f j < where 0 b n r 0 = r = b r k = (b k r k r k ) k = n and r = b n r n r n = ( ) n det(b) We would lke to make an observaton at ths pont: snce the nverse of a trdagonal matrx s a full matrx the Schur s complement method s not very effcent Moreover Vmur [] obtaned the nverse of another partcular trdagonal matrx n terms of the Gegenbauer polynomal C α n (x) (for α = ) whose generatng functon s ( xtt ) α = n=0 C α n (x)tn ; Prabhakar et al [3] showed some connecton between the aforementoned nverse and the generalzed Hermte polynomals namely g m n (x λ) = [n/m] k=0 n! k!(n mk)! λk x n mk (Classcal Hermte H n (x) are obtaned for m = λ = x x) Many specal cases of the banded matrces such as Toepltz matrces symmetrc Toepltz matrces especally trdagonal matrces etc have been studed by several authors as we prevously mentoned (the reader can fnd many more references publshed n the present journal for nstance [4] among others) These matrces arse n many areas of mathematcs and ts applcatons Trdagonal or more general banded matrces are used n telecommuncaton system analyss fnte dfference methods for solvng PDEs lnear recurrence systems wth nonconstant coeffcents etc so t s natural to ask the queston of whether one can obtan some results about the nverse of 4dagonal or perhaps even general banded matrces We wll do so n ths paper Throughout ths paper we consder a general rbanded matrx B rn of order n defned by a a a 3 a r 0 0 a a a a 3 a r a 3 a a 3 a 3 a B rn = a 3 a 3 a 4 a 4 a r n r b j = a r a 3 3 a 4 a 5 a 3 n 3 0 a r a n a 3 n a 3 n a n a n a n 0 0 a r n r a 3 n where a j s stand for arbtrary real numbers r r j r n (note that the n the notaton a n does not denote the th power of a n ) When r = n the matrx B nn s reduced to an arbtrary square matrx In ths paper generalzng a method of [5] we gve the LU factorzaton and (n our man result) the nverse of the matrx B rn (f t exsts) Our results are vald for an arbtrary square matrx (by takng r = n) Therefore we gve a new approach for computng the nverse of an nvertble square matrx thus generalzng varous results (see [6 8590] and the references theren) Our method s based on Hessenberg submatrces assocated to B rn Secton deals wth the LU factorzaton of an rbanded matrx To fnd the nverse of such a matrx (obtaned n Secton 4) we need to fnd nverses of the obtaned trangular matrces from the LU factorzaton and ths s done n Secton 3 We conclude the paper wth a few examples n Secton 5 a n a n () LU factorzaton of an rbanded matrx Ths secton s manly devoted to the LU factorzaton of the matrx B rn Frst we construct two recurrences For r and s r defne k = s a r s t= m t s t kt s t ()
3 8 E Kılıç P Stanca / Journal of Computatonal and Appled Mathematcs 37 (03) 6 35 and for r () as m = s r t= k s m t s t kt s t (3) wth ntal condtons and m = a () k k = a r = a () r a where the terms a ± n for n are the entres of B rn For these sequences to be welldefned we assume that none of the denomnators k s are zero (whch s equvalent to the belowdefned U and consequently B rn beng nvertble) Now defne untary lower trdagonal matrx L and upper trangular matrx U and L = l j = 0 m m m m m 3 m r 0 m r m 3 m 4 m 4 m n 0 0 m r n r m n m n k k k 3 k r 0 0 k k k 3 k r U = k 3 k 3 k u j = k 4 k 4 k r k 5 k 3 n 3 k n k n n r k 3 n k n 0 k n Our frst result gves the LU factorzaton of B rn (5) (4) Theorem For n > the LU factorzaton of matrx B rn s gven by B rn = LU where L and U are defned as n (4) and (5) respectvely Proof Frst we consder the case = j r From matrx multplcaton and the defntons of L and U we have b = n l s u s = s= l s u s s= = l u l u l u = m k m k m k k From () by takng r = we obtan b = a
4 E Kılıç P Stanca / Journal of Computatonal and Appled Mathematcs 37 (03) whch gves the concluson Now consder the case of > r Thus b = n l s u s = s= r l s u s s= = l u l u l r u r From the defntons of L and U we wrte b = k m k m k3 mr r kr r whch by takng = n () mples b = a n whch shows the clam for = j Next we look at the superdagonal entres of matrx B rn Consder the case j = q where q r Thus by the defnton of B n for q r b q = a q = n l s u sq s= We consder two cases Frst we assume r q and so l t u tq = l u q l u q l u q t= = m k q m q k q m kq kq whch by takng q and n n () gves us t= l t u tq = a q whch completes the proof for the frst case Now we consder the case r q < Thus we wrte b q = a q = n r q l s u sq = l t u tq s= t= From the defntons of matrces U and L we can wrte r q l t u tq = l u q l u q l r q u rqq t= Usng () we obtan b q = a q = k q m kq m k3q mr q rq kr rq r q = k q m t t kqt t t= whch completes the proof for the upper dagonal entres of matrx B rn Fnally we look at the upper dagonal entres of the matrx B rn and we need to show that b q = a (q) Here we frst consder the case r q From the defntons of matrces U and L b q = n l qt u t = t= l qt u t t= = m q k mq k m q k mq k = m q k t= m qt t kt t
5 30 E Kılıç P Stanca / Journal of Computatonal and Appled Mathematcs 37 (03) 6 35 whch by takng n and q n () mples b q = a (q) For the fnal case that s > r q usng the defntons of U and L we wrte b q = n r q l qt u t = l q t u t t= t= = l q u l q u l q rq u rq = m q k m q k mq(r q ) rq r q = m q k t= m qt t kt t whch by takng n and q n () gves b q = a (q) and the theorem s proved k r q rq The result of Theorem wll be vald for the LU factorzaton of any arbtrary square matrx by takng r = n n the matrx B rn Now we gve a closed formula for det B rn usng the LU factorzaton of Brn Corollary For n > 0 n det B rn = = k where k s are gven by () 3 The nverse of trangular matrces In ths secton we gve an explct formula for the nverse of a general trangular matrx For ths purpose we construct certan submatrces of a trangular matrx whch are Hessenberg matrces and then we consder the determnants of these submatrces to determne the entres of the nverse of the consdered trangular matrx Snce upper and lower trangular matrces have essentally the same propertes frst we consder the upper trangular matrx case We denote the correspondng Hessenberg matrces for an upper and a lower trangular matrx by H u (r s) and H l (r s) respectvely Let H = h j be an arbtrary (n n) upper trangular matrx Now we construct square Hessenberg submatrces of H hj of order s r n the followng way: for s > r > 0 let H u (r s) = denote an upper Hessenberg submatrx of H by deletng ts frst r and last (n s) columns and frst (r ) and last (n s ) rows Clearly the (s r) (s r) upper Hessenberg matrx H u (r s) takes the form: h rr h rr h rs h rs h rr h rr h rs h rs H u (r s) = 0 h s s h s s h s s 0 0 h s s h s s (6) Smlarly let H = h j be an arbtrary (n n) lower trangular matrx We construct square Hessenberg submatrces of ĥj H of order (r s) n the followng way: for r > s > 0 let H l (r s) = denote a lower Hessenberg submatrx of H by deletng ts frst r and last (n s) rows and frst (r ) and last (n s ) columns Clearly the (r s) (r s) lower Hessenberg matrx H l (r s) takes the form: h rr h rr 0 0 h rr h rr H l (r s) = hs s h s r h s r h s s h s s h sr h sr h ss h ss
6 E Kılıç P Stanca / Journal of Computatonal and Appled Mathematcs 37 (03) Here we note that (H u (r s)) T = (H l (s r)) Our constructon s vald for any upper or lower trangular matrx but we wll apply t only to L U gven by (4) and (5) renderng L l ( j) U u ( j) For example let A be an upper trangular matrx of order 6 as follows: a b c d e f 0 a b c d e A = 0 0 a 3 b 3 c 3 d a 4 b 4 c a 5 b a 6 Thus A u ( 5) and A u (3 6) take the forms : b c d b3 c 3 d 3 A u ( 5) = a 3 b 3 c 3 and A u (3 6) = a 4 b 4 c 4 0 a 4 b 4 0 a 5 b 5 Here note that all matrces of the form H u ( j) (H l ( j)) obtaned from an upper (lower) trangular matrx U are upper (lower) Hessenberg matrces Now we start wth the followng two lemmas Throughout ths paper we assume the boundary condtons H u (r r) = H l (r r) = and j x = for > j Lemma 3 Let the (j ) (j ) upper Hessenberg matrx H u ( j) be defned as n (6) Then for j > j j det H u ( j) = ( ) j t a j tj det (H u ( j t)) a kk t= k=j t Proof If we compute the determnant of the upper Hessenberg matrx H u ( j) by the Laplace expanson of a determnant wth respect to the last column then the proof follows Theorem 4 Let U = a j be an (n n) arbtrary upper trangular matrx and W = wj = U ts nverse Then (a ) f = j w j = j a kk ( ) j det (H u ( j)) j > k= where H u (r s) s as before Proof Denote WU by E = e j It s clear that for the case = j E = In where I n s the nth unt matrx Now consder the case j > From the defntons of matrces W and U e j = n w t a tj = t= = a j a j w t a tj t= j t a kk ( ) t a tj det (H u ( t)) t= k= = a j a ( ) a j det (H u ( )) a a ( ) a j det (H u ( )) a a a = ( )j a j j det (H u ( j )) j a kk j k= a kk a j ( ) k= j k= j a j j ( ) j ( ) j det (H u ( j)) k=j a kk a j ( ) ( ) j a j j det (H u ( j )) ( ) j det (H u ( j)) j j a kk j k= k= a kk det (H u ( )) det (H u ( j )) ( ) j a j j det (H u ( j )) a kk k=
7 3 E Kılıç P Stanca / Journal of Computatonal and Appled Mathematcs 37 (03) 6 35 = j j j a kk k= t= k=t a kk ( ) t a t j det (H u ( t )) whch by Lemma 3 mples e j = 0 and the proof s completed ( ) j det (H u ( j)) All the results of ths secton hold for lower trangular matrces wth the obvous modfcaton usng the lower Hessenberg matrx Now we menton an nterestng fact that the numbers of summed or subtracted terms n computng the nverse of a term of an upper (lower) trangular matrx are the generalzed orderk Fbonacc numbers defned by f k n = k = c f k n n > 0 where f k = f k = = k f k = 0 0 When k = and c = c = the generalzed order Fbonacc numbers are the usual Fbonacc numbers that s f = m F m (mth Fbonacc number) When also k = 3 c = c = c 3 = then the generalzed order3 Fbonacc numbers by the ntal condtons f 3 = f 3 = 0 f 3 = 0 are whch are also known as trbonacc numbers Let U n = u j be an upper (wth k superdagonals) trangular matrx of order n wth u = h for n u r = cr t= a r for n r r k and h s are all dstnct from zero and a r s are arbtrary For computng the nverse of U n we need the correspondng Hessenberg submatrces defned as before by H (r s) Therefore we should note that the numbers of summed or subtracted terms n computng the nverse of a term of U n omttng the sgns and denomnators of the terms that s the number of requred summatons n the expanson of det (H (r s)) are the generalzed order(s r) Fbonacc numbers f s r n To show that we consder a (k k) upper Hessenberg matrx H k = a j wth rsuperdagonals whose entres are gven by a = e for k a r = c r t= h(r) t for k r and 0 r k e 0 for all If one superdagonal has a 0 entry then all the entres n ths superdagonal are zeros That s f a r = 0 for some and r then c r = 0 Here E n denotes the number of summed or subtracted terms n det H n For example f a b c H 3 = d e f 0 g h then det H 3 = ahe afg bdh cdg and so the correspondng E 3 = 4 By expandng det H n wth respect to the frst row wthout any smplfcaton n the entres h (r) = c r t= a r we get E n = c E n c E n c r E n r One easly computes that E = E = E 3 = 4 E r = r Consequently one can see that E n = f k n where f k n s the generalzed orderk Fbonacc numbers For example for k = 4 r = let c = and c = that s a = t= h() t = h () h () a = and so h () h() h () 0 0 e H 4 = h () h() h () 0 0 e h () 3 h() 3 h () e 3 h () 4 h() 4 Snce r = c = c = the countng sequence E n satsfes E n = E n E n wth E = E = so E n = P n the well known Pell sequence Therefore the number of summed or subtracted terms whle computng det H 4 s the 5th Pell number: h () h() h () 0 0 e h () det h() h () 0 0 e h () 3 h() 3 h () e 3 h () 4 h() 4 = e e 3 h h 3 h h h 3 h 4 h h h 3 h 4 h h h 4 h 3 h h 3 h h 4 t= h() t = h ()
8 E Kılıç P Stanca / Journal of Computatonal and Appled Mathematcs 37 (03) h h h 3 h 4 h h h 3 h 4 h h 3 h h 4 h h h 4 h 3 h h h 3 h 4 h h h 4 h 3 h h 3 h h 4 h h h 3 h 4 h h h 3 h 4 h h 3 h h 4 h h h 4 h 3 h h h 3 h 4 e h h 3h 4 e h h h 4 e 3 h h h 3 e h h 3h 4 e h h 4h 3 e h h h 4 e h h h 4 e 3 h h 3 h e 3 h h h 3 e h h 3h 4 e h h h 4 e 3 h h 3 h From the above example t s seen that there are 9 terms n the expanson of det H 4 whch s the 5th Pell number P 5 4 The nverse of an rbanded matrx In ths secton we gve a closed formula for the nverse of an rbanded matrx Frst we consder the nverse of the upper trangular matrx L Here we recall a well known fact that the nverse of an upper trangular matrx s also upper trangular We shall gve the followng lemmas whose proofs are straghtforward Lemma 5 Let the lower trangular matrx L be as n (4) Let E = e j denote the nverse of L Then f = j e j = ( ) j det (L l ( j)) f j < where L l ( j) s defned as before Lemma 6 Let the upper trangular matrx U be as n (5) Let G = g j denote the nverse of U Then f j = k g j = ( ) j det (U u ( j)) f j > j k r r= where U u ( j) s defned as before The man result of ths secton follows from the LU factorzaton of B rn and the prevous lemmas Theorem 7 Let D n = d j denote the nverse of the matrx Brn Then d j = n t= g te tj where g t e tj are defned n the prevous lemmas Precsely f we let S( t j) = det(u u(j)) det(l l (tj)) j then r= k r ( ) j det(u u( j)) n ( ) j j S( t j) d j = k k r r= n S( t ) t> ( ) j det(l l ( j)) ( ) j k t>j f < j f = j n S( t j) f > j t> Proof Snce B rn = LU then by the prevous two lemmas we get D = U L and so n n d j = g t e tj = g t e tj t= t=max{j} By takng the three cases < j = j > j and usng the expressons of g t e tj from Lemmas 5 and 6 the clam follows 5 Some partcular cases If we take r = and S n to be a symmetrc matrx and label a = a b = b = b we obtan the formulas of [] for the nverse of
9 34 E Kılıç P Stanca / Journal of Computatonal and Appled Mathematcs 37 (03) 6 35 a b 0 b a b b 3 a 3 b 4 0 S n = 0 b n a n b n 0 0 b n a n namely the exstence of two sequences u v such that u v u v u v 3 u v n u v u v u v 3 u v n S = u v 3 u v 3 u 3 v 3 u 3 v n n u v n u v n u 3 v n u n v n The sequences u v j can be determned (as Meurant dd n []) usng the LU decomposton of Theorem and they wll depend on our k j mj We wll not repeat the argument here Consder the general bnary recurrence G n = a G n b G n G 0 = 0 G = Let 0 0 a 0 0 b a 0 0 G 3n = 0 a 0 0 b a Its nverse s [] G 0 0 G G 0 0 G 3 G G 0 0 G 3n G n G n G G 0 G n G n G 3 G G In the same manner by gong through our argument one can obtan many other known (and possbly unknown) matrx nverses and determnants We end by dsplayng yet another example based on the Chebyshev polynomals of the second knd U n (x) = sn(n)θ cos θ = x whch satsfy the recurrence: U sn θ n (x) = xu n (x) U n (x) U 0 (x) = a b 0 0 b a b 0 U (x) = x The symmetrc Toepltz S = has the nverse [3] S = (t j ) j where t j = 0 b a b 0 b a U (a/b)u n j (a/b) ( ) j j b U n (a/b) ( ) j U j (a/b)u n (a/b) > j b U n (a/b) It could be nterestng to use a varaton of the methods of ths paper to nvestgate the spectrum of general rbanded matrces (see [4] for example) Acknowledgments The authors thank the referee for pontng out several shortcomngs n the presentaton of the paper The second author was partally supported by Ar Force CVAQ (D Nussbaum) and NPSRIP References [] P Schlegel The explct nverse of a trdagonal matrx Math Comp 4 (970) 665 [] V Vmur A novel way of expressng a class of trdagonal band matrx n terms of Gegenbauer polynomals Matrx & Tensors Quart 0 (969) [3] TR Prabhakar A Srvastava RC Tomar Inverse of a trdagonal band matrx n terms of generalzed Hermte polynomals Indan J Pure Appl Math 0 (979) 84 87
10 E Kılıç P Stanca / Journal of Computatonal and Appled Mathematcs 37 (03) [4] TS Papatheodorou Inverses for a class of banded matrces and applcatons to pecewse cubc approxmaton J Comput Appl Math 8 (4) (98) [5] RK Mallk The nverse of a lower trangular matrx Int J Math Game Theory Algebra 8 (999) [6] CM da Fonseca J Petronlho Explct nverses of some trdagonal matrces Lnear Algebra Appl 35 (00) 7 [7] E Klc Explct formula for the nverse of a trdagonal matrx by backward contnued fractons Appl Math Comput 97 (008) [8] HB L TZ Huang XP Lu H L On the nverses of general trdagonal matrces Lnear Algebra Appl 433 (5) (00) [9] RK Mallk The nverse of a trdagonal matrx Lnear Algebra Appl 35 (00) [0] M ElMkkawy A Karawa Inverson of general trdagonal matrces Appl Math Lett 9 (006) 7 70 [] G Meurant A revew on the nverse of symmetrc trdagonal and block trdagonal matrces SIAM J Matrx Anal Appl 3 (99) [] W Wang T Wang Identtes va Bell matrx and Fbonacc matrx Dscrete Appl Math 56 (008) [3] U Kamps Chebyshev polynomals and least squares estmaton based on onedependent random varables Lnear Algebra Appl (989) 7 30 [4] A Böttcher SM Grudsky EA Maksmenko Insde the egenvalues of certan Hermtan Toepltz band matrces J Comput Appl Math 33 (00) 45 64
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