EXPLICIT INVERSE OF A TRIDIAGONAL k TOEPLITZ MATRIX

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1 EXPLICIT INVERSE OF A TRIDIAGONAL k TOEPLITZ MATRIX C M DA FONSECA AND J PETRONILHO Abstract We obtai explicit formulas for the etries of the iverse of a osigular ad irreducible tridiagoal k Toeplitz matrix A The proof is based o results from the theory of orthogoal polyomials ad it is show that the etries of the iverse of such a matrix are give i terms of Chebyshev polyomials of the secod kid We also compute the characteristic polyomial of A which eable us to state some coditios for the existece of A 1 Our results exteds some other oes i the literature kow for the case whe the residue mod k of the order of A equals 0 or k 1 1 Itroductio Tridiagoal matrices arise i may cotexts i pure ad applied mathematics For istace, besides their ow iterest i liear algebra, they are a basic tool i approximatio theory, particularly i the study of special fuctios ad orthogoal polyomials They also arise aturally i umerical aalysis ad partial differetial equatios, i the discretizatio of elliptic or parabolic partial differetial equatios by fiite differece methods For applicatio of such methods, the study of the iverse of the ivolved matrices appears to be very importat For a review of the iverse of a tridiagoal ad block tridiagoal matrix, as well as applicatios of tridiagoal ad iverse of tridiagoal matrices, we refer the reader to the paper by G Meurat [16] as well as the refereces therei As occur i may problems i mathematics, closed explicit formulas to express the quatities which appear i the problems i study ca be give oly i some special situatios, ad this is certaily the case for the elemets of the iverse of a geeral tridiagoal matrix As referred i [16], it seems that D Moskovitz [17] was the first to give a explicit formula for the iverse of a tridiagoal matrix i coectio with oe dimesioal ad two dimesioal Poisso problems Further, a closed explicit formula for the iverse is kow from several refereces i the case of a tridiagoal Toeplitz matrix cf P Schlegel [19], T Yamamoto ad Y Ikebe [21], Heiig ad Rost [9, p28], eg, ie, a tridiagoal matrix with costat etries alog the diagoals parallel to the mai diagoal, which is a matrix that correspods to a discretizatio of a elliptic partial differetial equatio with costat coefficiets cf 51 bellow I [5] the authors also gave closed explicit formulas for the iverse of a irreducible tridiagoal matrix with 2 periodic ad 3 periodic etries alog the diagoals parallel to the mai diagoals Other explicit examples for the etries of the iverse of some Date: November 19, Mathematics Subject Classificatio 15A09, 42C05, 33C45, 65Q05 Key words ad phrases tridiagoal k Toeplitz matrices, iversio of matrices, orthogoal polyomials, recurrece relatios This work was supported by CMUC Cetro de Matemática da Uiv Coimbra 1

2 2 C M DA FONSECA AND J PETRONILHO tridiagoal matrices ca be foud i Lewis [11] Formulas usually ot explicit for the iverse of a geeral tridiagoal matrix, as well as criteria for a give matrix to have a tridiagoal iverse, also have bee give by several authors, some of them ivolvig recurrece relatios or some caoical decompositios for the matrices i cosideratio for details, see agai [16] ad the refereces therei, eg I this paper we geeralize the above metioed formula for the iverse of a tridiagoal Toeplitz ad tridiagoal 2 periodic ad 3 periodic matrices to some more geeral matrices I fact, the purpose of this paper is to calculate explicitly the elemets of the iverse of a tridiagoal k Toeplitz matrix, a cocept itroduced by M J C Gover ad S Barett [8] By defiitio, give a matrix A =[a ij ]of order N ad a positive iteger umber k N, A is a tridiagoal k Toeplitz matrix if a i+k,j+k = a ij i, j =1, 2,,N k This meas that A is a tridiagoal matrix of the form a 1 b 1 c 1 ak b k c k a 1 b 1 11 A = c 1 C N N ak b k c k a 1 b 1 c 1 Notice that if k = 1 the A is a tridiagoal Toeplitz matrix; ad whe the order N is a multiple of k the A is a block Toeplitz matrix If b i c i 0 for all i =1,,k we say that A is irreducible I this paper we will assume that A is irreducible The structure of the paper is as follows I the ext sectio some basic tools cocerig the geeral theory of orthogoal polyomials are preseted I sectio 3 we cosider systems of orthogoal polyomials that ca be obtaied from other old oes via a polyomial mappig, i some appropriate way describe further o, ad we obtai explicit formulas, i terms of the old polyomials, for the ew polyomials obtaied by the polyomial mappig I sectio 4 a review of some kow facts cocerig the iverse of a tridiagoal matrix is give, poitig out the coectio with the theory of orthogoal polyomial As applicatio of the results obtaied i the previous sectios, i sectio 5 we give explicit expressios for the etries of the iverse of the tridiagoal k Toeplitz matrix A Essetially we show that the etries of this iverse ca be computed by usig a appropriate polyomial mappig takig as old polyomials the classical Chebyshev polyomials of the secod kid The results preseted i sectio 5 are stated assumig that the matrix A is osigular ad so i sectio 6 we will state coditios for the existece of A 1 These coditios will be obtaied from a explicit expressio for the characteristic polyomial of A Such expressio geeralizes results by L Elser ad R M Redheffer [4] stated for the case whe the residue mod k of the order of A equals 0 or k 1

3 EXPLICIT INVERSE OF A TRIDIAGONAL k TOEPLITZ MATRIX 3 2 Prelimiary results o orthogoal polyomials Let P be the complex vector space C[x] adp be its algebraic dual A sequece P } 0 Pis called a orthogoal polyomial sequece OPS if each P has degree exactly for every =0, 1, 2, it is assumed that P 0 x is a o-zero costat ad there exists a liear fuctioal L P such that L,P P m = k δ,m k 0,,m =0, 1, 2, where L,f meas the actio of the fuctioal L over the polyomial f ad δ,m deotes the usual Kroecker symbol I such a case the liear fuctioal L is uique, ad we say that P } 0 is a OPS with respect to L The theoretical basis for most orthogoality proofs i this paper is the so called Favard s theorem, which states that ay OPS P } 0 is characterized by a three-term recurrece relatio 21 xp x =α P +1 x+β P x+γ P 1 x, =0, 1, 2, with iitial coditios P 1 x =0adP 0 x = cost 0, where α } 0, β } 0 ad γ } 0 are sequeces of complex umbers such that α γ +1 0 for all =0, 1, 2, To be more exact, although this theorem is usually attributed to J Favard, it is kow sice T J Stieltjes, amely i the positive-defiite case, ie, whe the momet liear fuctioal L is characterized by some positive distributio fuctio σ with fiite momets such that L,f := + fxdσx, f P, the itegral beig i the Stieltjes sese If P } 0 ad Q } 0 are two OPS s with respect to the same liear fuctioal L the there exists a sequece c } 0 of o-zero complex umbers such that Q x =c P x, =0, 1, 2, Hece we ca assume α =1 =0, 1, 2, i the above three-term recurrece relatio, ie, i geeral we will deal with moic orthogoal polyomial sequeces MOPS I matrix form the three-term recurrece relatio 21 ca be writte as x P 0 x P 1 x P x = J +1 P 0 x P 1 x P x + α P +1 x 0 0 1, where J +1 is a tridiagoal Jacobi matrix of order + 1, defied by β 0 α γ 1 β 1 α γ 2 β J +1 := =0, 1, 2, β 1 α γ β I fact, usually a Jacobi matrix is uderstaded as a symmetric tridiagoal matrix, but here we avoid this distictio It follows that if x j } j=1 is the set of zeros of the polyomial P the each x j is a eigevalue of the correspodig Jacobi matrix

4 4 C M DA FONSECA AND J PETRONILHO J of order, ad a associated eigevector is [ P 0 x j,p 1 x j,,p 1 x j ] t Moreover, the moic characteristic polyomial of J is precisely P, ie, P x = det xi J, =1, 2,, where I deotes the idetity matrix of order Two of the most useful OPS s are the Chebyshev polyomials of the first ad secod kid, deoted by T } 0 ad U } 0, respectively These polyomials satisfy the three-term recurrece relatios xt x =T +1 x+t 1 x, T 0 x =1, T 1 x =x, 22 xu x =U +1 x+u 1 x, U 0 x =1, U 1 x =2x for all =1, 2, It is well kow cf [3], eg that U ad T also satisfy si +1θ T x =cosθ, U x =, x =cosθ 0 θ<π si θ for all =0, 1, 2 where, if si θ =0,si +1θ/ si θ must be replaced by its limit as θ 0, from which oe easily deduce the orthogoality relatios 1 dx T xt m x = π if = m =0 1 1 x 2 π 2 δ,m otherwise, ad 1 1 U xu m x 1 x 2 dx = π 2 δ,m We also otice that the MOPS s T } 0 ad Û} 0, correspodig to T } 0 ad U } 0 resp, are give by 23 Û x =2 U x, T x =2 1 T x for all =0, 1, 2 Fially, we will make use of the followig asymptotic result: 24 U z lim + U 1 z = z + z 2 1, where the square root is take i such a way that z+ z 2 1 is a aalytic fuctio i C \ [ 1, 1] with z + z 2 1 > 1 whe z [ 1, 1] 3 Descriptio of the polyomial mappig Accordig to the Favard theorem, ay give MOPS p x} 0 ca be characterized by a three-term recurrece relatio For our purposes, it is coveiet to write this recurrece as a geeral block of recurrece relatios of the type x b j p k+j x =p k+j+1 x+a j p k+j 1 x, 31 j =0, 1,,k 1; =0, 1, 2,, ad satisfyig iitial coditios p 1 x =0, p 0 x =1 Without lost of geerality, we will take a 0 0 = 1, ad polyomials p s with degree s 1 will be always defied as the zero polyomial Also, we make the covetio that empty sum equals zero, ad empty product equals oe Sice we assume that

5 EXPLICIT INVERSE OF A TRIDIAGONAL k TOEPLITZ MATRIX 5 p x} 0 give by 31 is a sequece of orthogoal polyomials, we eed to impose the coditios a j 0, j =0, 1,,k 1; =0, 1, 2, We also assume that k 3 Followig J Charris, M E H Ismail ad S Mosalve [1], for =0, 1, 2,, we itroduce polyomials i, j; by 0 if j<i 2 i, j; x := 1 if j = i 2 x b i 1 if j = i 1 ad, for j i 1, by the determiatal form x b i a i x b i a i+1 x b i i, j; x := x b j a j x b j I [1] the authors showed that i order to determie the polyomials p for all =0, 1, 2, it is oly eed to compute the p k s for all =0, 1, 2 A special case of mai iterest i applicatios occurs whe the give MOPS p } 0 is obtaied from aother MOPS q } 0 via a polyomial mappig, i the sese that there exists a polyomial T of degree exactly k such that, up to a affie chage i the variables, 32 p k x =q T x, =0, 1, 2, Assurace of the existece of such a orthogoal sequece q } 0 is ot easy i practice It is kow, eg, that if p } 0 is obtaied from some other system of orthogoal polyomials via a polyomial mappig i the above sese the b 0 ad 2,k 1; x must be idepedet of see [1],[6] This, however, is ot a sufficiet coditio, as examples show A sufficiet coditio has bee improved i [1] cf Theorem 41 ad Remark 42, ad it states that if both b 0 ad 2,k 1; x are idepedet of ad if, i additio, a 0 1 2,k 2; x+a 1 3,k 1; x a ,k 1; x is idepedet of x for every =1, 2,, the p } 0 ca be obtaied via a polyomial mappig of the type 32 These kid of polyomial mappigs, such as 32, were extesively studied by J Geroimo ad W Va-Assche i [6] ad J Charris ad M E H Ismail i [2] Some examples, makig a coectio with the so-called sieved orthogoal polyomials o the real lie ad o the uit circle, were give i [10] ad [18] Other examples, i the particular cases of quadratic ad cubic polyomial mappigs, ca be foud i [13], [14] ad [15] A polyomial mappig of the type 32 comes essetially from the expasio of the p k+j s i terms of p k ad p +1k I the ext we will cosider a slightly differet polyomial mappig, which we foud to be more appropriate to study

6 6 C M DA FONSECA AND J PETRONILHO tridiagoal k Toeplitz matrices The idea cosists of expadig the p k+j s i terms of p k 1 ad p +1k 1, which will lead to a polyomial mappig of the type p k+k 1 x =ρxq T x, =0, 1, 2,, where ρ ad T are fixed polyomials of degree k 1adk, respectively This fact is stated i the ext propositio Theorem 31 Let p } 0 be a MOPS characterized by the geeral block of recurrece relatios 31 For =0, 1, 2,, defie r x :=a ,k 2; x a ,k 2; x + a k 1 1,k 3; x a k ,k 3; x Assume that, for all =0, 1, 2,, the followig coditios hold: i b k 1 is idepedet of ; ii 1,k 2; x =:ρx is idepedet of for every x ; iii r x =:r is idepedet of x for every Cosider the polyomial T of degree k defied as T x := 0 1,k 1; x a ,k 2; x, ad let q } 0 be the MOPS geerated by the recurrece relatio 33 q +1 x =x r q x s q 1 x, =0, 1, 2, with iitial coditios q 1 x =0 ad q 0 x =1,where s := a 0 a 1 a k 1 The, for each j =0, 1, 2,,k 1 ad all =0, 1, 2, 34 p k+j x = 1,j 1; x q T x + a 0 a 1 a j j +2,k 2; x q 1 T x I particular, for j = k 1, 35 p k+k 1 x =ρx q T x, =0, 1, 2, Proof We begi by rewritig 31 as a system i matrix form, V p k p k+1 p k+k 3 p k+k 2 p +1k = a 0 p k p +1k 1 x b k 1 p +1k 1,

7 EXPLICIT INVERSE OF A TRIDIAGONAL k TOEPLITZ MATRIX 7 where V is the tridiagoal matrix of order k defied by V := x b 0 1 a 1 x b 1 1 a 2 x b 2 1 a k 3 x b k 3 a k 2 1 x b k 2 0 a k 1 1 Solvig this system for p k+j i terms of p k+k 1 ad p k 1 by Cramer s rule, it follows that the polyomials of the sequece p } 0 satisfy the relatios 36 1,k 2; xp k+j x ad = 1,j 1; xp +1k 1 x+a 0 a 1 a j j +2,k 2; xp k 1 x, j =0, 1,,k 2, =0, 1, ,k 2; xp k+k x } = x b k 1 1,k 2; x a k 1 1,k 3; x p +1k 1 x a 0 a 1 a k 1 p k 1 x, =0, 1, 2 Hece, oe sees that i order to determie the polyomials p i for all i =0, 1, 2, we oly eed to compute the p k 1 s for all =0, 1, 2 I 36 replace by + 1 ad the set j = 0 to fid ,k 2; xp k+k x = p +2k 1 x+a ,k 2; xp +1k 1 x for every =0, 1, 2 Sice 2,k 1; x is idepedet of, the left-had sides of 37 ad 38 coicide, so that after a ew chage of idices 1, 39 p +1k 1 x+a 0 1 a1 1 ak 1 1 p 1k 1x = x b k ,k 2; x a k ,k 3; x } a 0 2,k 2; x p k 1 x, =1, 2 Now, expasio of the determiat 0 1,k 1; x alog its last row gives 0 1,k 1; x =x b k ,k 2; x a k ,k 3; x

8 8 C M DA FONSECA AND J PETRONILHO Therefore, 39 ca be rewritte as p +1k 1 x+a 0 1 a1 1 ak 1 1 p 1k 1x = 0 1,k 1; x a k ,k 3; x a k ,k 3; x } p k 1 x = T x r 1 x } p k 1 x for all =1, 2, hece p +1k 1 x =T x r 1 p k 1 x s 1 p 1k 1 x, =1, 2 Sice p k 1 x = 0 1,k 2; x ρx, 35 comes ow easily by iductio over Fially, 34 is a immediate cosequece of 36 ad 35 4 Iverse of a tridiagoal matrix Let us cosider a geeral tridiagoal matrix of order N, say β 1 α γ 1 β 2 α γ 2 β J = β N 1 α N γ N 1 β N Whe J is ivertible, J 1 ca be computed accordig to the followig Propositio 41 Usmai [20] Assume that J is ivertible The the etries of J 1 are give by 42 1 i+j α J 1 i α j 1 θ i 1 φ j+1 /θ N if i j ij = 1 i+j γ j γ i 1 θ j 1 φ i+1 /θ N if i>j, where 43 θ 1 =0,θ 0 =1,θ = β θ 1 α 1 γ 1 θ 2 =1, 2,,N, ad 44 φ N+2 =0,φ N+1 =1,φ = β φ +1 α γ φ +2 = N,N 1,,1 Heceforth, oe sees that the problem of the determiatio of the iverse of a osigular tridiagoal matrix reduces to solvig the differece equatios 43 ad 44 Sice those are homogeeous liear differece equatios of secod order with variable coefficiets, we ca ot expect to solve them explicitly uless for some special cases Notice that the above differece equatios also make clear a coectio betwee the theory of orthogoal polyomials ad the problem of the iversio of a tridiagoal matrix This coectio have bee explored by the authors i [5] to evaluate the iverse of tridiagoal 2 Toeplitz ad 3 Toeplitz matrices We poit out that the techique used i [5] is ot easy to apply to the geeral case of the tridiagoal k Toeplitz matrix, sice it does ot give a geeral procedure to solve the metioed

9 EXPLICIT INVERSE OF A TRIDIAGONAL k TOEPLITZ MATRIX 9 differece equatios A procedure to solve a geeral secod order differece equatio have bee preseted recetly by R K Mallik [12], where the author applies the results to determie the iverse of some matrices As will be show i the ext sectios, by usig the results preseted i sectio 3, the differece equatios ca be solved for some classes of tridiagoal matrices that iclude the tridiagoal k Toeplitz matrices 5 Iverse of a tridiagoal k Toeplitz matrix Explicit formulas for the iverse of a osigular tridiagoal k Toeplitz matrix have bee give i the case k = 1 by several authors see some refereces i the itroductio I this case A is a tridiagoal Toeplitz matrix, a b c a b A = c a, b c a ad so, if A is osigular ad irreducible, puttig d := a/2 bc, the iverse is give by 1 i+j b j i U i 1 d U j d bc j i+1 if i j U d 51 A 1 ij = 1 i+j c i j U j 1 d U i d bc i j+1 if i>j U d Whe k = 2 the iverse of the assumed irreducible ad o-sigular tridiagoal 2 Toeplitz matrix 11 have bee give i [5]: if we put µ 2 := b 1 b 2 c 1 c 2, β := µ/b 1 c 1, ξ 2 x := 1 2µ x a 1x a 2 b 1 c 1 b 2 c 2 } we fix µ as oe square root of b 1 b 2 c 1 c 2, ad defie Q i ; α, γ} i 0 the sequece of moic polyomials such that Q 2i+1 x; α, γ :=x α µ i U i ξ 2 x, Q 2i x; α, γ :=µ i U i ξ 2 x + γu i 1 ξ 2 x}, where α ad γ are some parameters, the A 1 ij = 1 i+j b j i/2 p i b j i+1/2 q i θ i 1 φ j+1 /θ, if i j 1 i+j c j i/2 p j cq j i+1/2 j θ j 1 φ i+1 /θ, if i>j where p l =3 1 l /2,q l =3+ 1 l /2, z deotes the greater iteger less or equal to the real umber z, θ i = 1 i Q i 0; a 1,β,

10 10 C M DA FONSECA AND J PETRONILHO ad 1 i Q +1 i 0; a 1, 1/β if is odd φ i = 1 i+1 Q +1 i 0; a 2,β if is eve The iverse of a tridiagoal 3 Toeplitz matrix case k = 3 also have bee give i [5], but we poit out that a misprit appeared i the statemet of the correspodig theorem, amely i the defiitio of the polyomial P i If fact, the correct defiitio i [5, Theorem 51] must be a b c P i α β γ ; x := αβγ i Ui 1 2 αβγ [ ] a b c π 3 α β γ ; x αβ + γ Nevertheless, the ext procedure is true for ay k 3 I order to give the iverse of the geeral tridiagoal k Toeplitz matrix 11, with k 3, we eed to itroduce some otatio We deote z1,,z π k k ; x w 1,,w k x + z w 1 x + z w 2 x + z :=, x + z k w k 2 x + z k 1 1 w k w k 1 x + z k ad z1,,z s w 1,,w s 1 := ; x x + z w 1 x + z w 2 x + z x + z s w s 1 x + z s, so that π k is a moic polyomial of degree exactly k i x, depedig o 2k give complex parameters, ad is a moic polyomial of degree s i x which depeds o 2s 1 give parameters with the usual covetios 0ifs<0, 1if s =0ad x + z 1 if s = 1 The, if w i 0 holds for every i, puttig w 2 := k i=1 w i

11 EXPLICIT INVERSE OF A TRIDIAGONAL k TOEPLITZ MATRIX 11 so that we choose w to be a square root of k i=1 w i, we set z1,,z U k,k ; x w 1,,w k := w 1 U π k 2w z1,,z k w 1,,w k ; x + 1 k w k + w 2 } /w k ad otice that U,k is a moic polyomial of degree k i x Fially, defie a sequece of moic polyomials Q i } i 0 by z1,,z Q k k+j ; x w 1,,w k z1,,z := j z1,,z ; x U k w 1,,w,k ; x j 1 w 1,,w k zj+2,,z + w k w 1 w j k 1 w j+2,,w k 2 ; x U 1,k z1,,z k w 1,,w k ; x, for =0, 1, 2, ad 0 j k 1, each Q i beig a moic polyomial of degree exactly i for every i With these otatios we ca state the followig Theorem 51 Let A be the tridiagoal k Toeplitz matrix of order N defied by 11, withn k 3 Assume that A is o-sigular ad irreducible The 1 i+j α i α i+1 α j θ i 1 φ j+1 /θ N if i j A 1 ij = 1 i+j γ j γ j+1 γ i θ j 1 φ i+1 /θ N if i>j, where α kl+s+1 = b s+1, γ kl+s+1 = c s+1 s =0,,k 1; l =0,, N s 1 k, ad the θ s ad the φ s are explicitly give by a1,,a θ = Q k ;0 ad, r beig a iteger umber characterized by 0 r k 1 ad N r mod k, a φ = Q r,a r 1,,a 1,a k,a k 1,,a r+1 N+1 ;0, b σr c σr,,b σ1 c σ1,b σk c σk,,b σr+1 c σr+1 where σ stads for the followig cycle of legth k: σ := r, r 1,, 1,k,k 1,,r+1 1 Remark 52 As remarked before, the case k = 2 have bee solved i [5] by usig a appropriate quadratic trasformatio However, the iverse of a tridiagoal 2 Toeplitz matrix also follows from the case k = 4 i the result above, whe we take a 3 = a 1, a 4 = a 2, b 3 = b 1, b 4 = b 2, c 3 = c 1 ad c 4 = c 2 Of course, whe all a i s are equal as well as all the b i s ad all the c i s, we also get the iverse of a tridiagoal Toeplitz matrix case k = 1 1 with σ := k, k 1,, 1 if r =0

12 12 C M DA FONSECA AND J PETRONILHO Proof of Theorem 51 Accordig to the cosideratios i the previous sectio, the problem of the determiatio of the iverse of the geeral tridiagoal k Toeplitz matrix 11 reduces to the determiatio of the trasformatios θ ad φ from I order to evaluate θ, otice that for the matrix 11 relatios 43 become the followig system of differece equatios θ k+j+1 = a j+1 θ k+j b j c j θ k+j 1, j =0, 1,,k 1; =0, 1, 2,, N j 1/k, with iitial coditios θ 1 =0 ad θ 0 = 1 Hece, oe sees that θ = p 0, =0, 1,,N, where p } 0 is the MOPS give by the block of recurrece relatios 31 with a j } ad b j } defied by a 0 = b k c k, a j = b j c j j =1, 2,,k 1, 52 b j = a j+1 j =0, 1,,k 1 Uder this coditios, it is clear that all the determiats i, j; x are idepedet of ad r x 0 is idepedet of x, so that all the hypothesis i-iii of Theorem 31 are fulfilled Moreover, r 0 = r =0, s =cost= k b i c i =: µ 2, =1, 2,, ad it follows that i this case the sequece q } 0 defied i Theorem 31 is explicitly give i terms of the Chebychev polyomials of the secod kid by q x =µ x U, =0, 1, 2, 2µ compare with 22 ad 23 Now, the polyomial T i Theorem 31 is give by 53 T x = 0 1,k 1; x b k c k 0 2,k 2; x, ad by some computatios o the determiat bellow oe easily verifies that T x+ 1 k+1 b k c k + k 1 i=1 b ic i x + a b 1 c 1 x + a b 2 c 2 x + a = x + a k 1 1 b k c k 0 0 b k 1 c k 1 x + a k a1,,a = π k k ; x otice that the first equality is true sice k 3 It follows that a1,,a q T x = U k,k ; x, =0, 1, 2, i=1

13 EXPLICIT INVERSE OF A TRIDIAGONAL k TOEPLITZ MATRIX 13 Further, otice that i this case we have 1,j 2; x = j +2,k 2; x = a 1,,a j b 1 c 1,,b j 1 c j 1 a j+2,,a k 1 b j+2 c j+2,,b k 2 c k 2 ; x, ; x As a cosequece, from 34 we coclude that a1,,a 55 p x =Q k ; x, =0, 1, 2, Hece the represetatio for θ as i the theorem is proved I order to compute the φ s, otice first that 44 gives φ k+j = a j+1 φ k+j+1 b j c j φ k+j+2, 56 j =0, 1,,k 1; =0, 1, 2,, N j/k, with iitial coditios φ N+2 =0 ad φ N+1 = 1 To fid the solutio of 56 it is coveiet to make a chage of variables Set ψ := φ N+1, =1, 2,,N Let r be the iteger umber characterized by 0 r k 1, N r mod k The by straightforward computatios from 56 we obtai ψ k+j+1 = b j ψ k+j a j ψ k+j 1, j =0, 1,,k 1; =0, 1, 2,, N j/k, with iitial coditios ψ 1 =0 ad ψ 0 = 1, where br j c 57 a j r j if 0 j<r := b k+r j c k+r j if r j k 1, ad ar j if 0 j<r 58 b j := a k+r j if r j k 1 Therefore, we see that ψ = p 0, =0, 1, 2,,N, where, ow, p } 0 is the MOPS give by the block of recurrece relatios 31 with a j } ad b j } defied by 57 ad 58 Hece, if we proceed mutatis mutadis exactly as we have doe above for the determiatio of the θ s, for this sequece p } 0 we fid p x =Q a r,a r 1,,a 1,a k,a k 1,,a r+1 b σr c σr,,b σ1 c σ1,b σk c σk,,b σr+1 c σr+1 ; x, where σ := r, r 1,, 1,k,k 1,,r+ 1 is a cycle of legth k For x =0 this gives ψ Hece we get the desired expressio for φ

14 14 C M DA FONSECA AND J PETRONILHO 6 Some remarks o ivertibility coditios Our aim, i this sectio, is to give some iformatio about ivertibility coditios for the k Toeplitz matrix A I what follows we put b := k i=1 b i, c := k i=1 c i, µ 2 := bc For j>i, defie x a i b i c i x a i b i+1 c i+1 x a i i,j x :=, x a j b j 1 c j 1 x a j so that i,j is a polyomial of degree j i +1ix, ad for j i set 0 if j<i 1 i,j x := 1 if j = i 1 x a i if j = i We also defie the followig polyomial of degree k ϕ k x := 1k 2µ } D k x+b k c k + µ 2 /b k c k where D k is a moic polyomial of degree k, x a b 1 c 1 x a b 2 c 2 x a D k x := x a k 1 1 b k c k 0 0 b k 1 c k 1 x a k It is importat to keep i mid that i what follows we cosider that these polyomials ϕ k ad D k are defied oly whe k 3 We begi by poitig out that, up to a costat factor the characteristic polyomial for A, px; A := detxi N A, I N beig the idetity matrix of order N, is obtaied by a affie chage i the variable i the polyomial Q N, itroduced i the previous sectio, correspodig to the etries which appear i A Theorem 61 Assume that the tridiagoal k Toeplitz matrix A of order N is irreducible, with k 3 The its characteristic polyomial is px; A = 1 N a1,,a Q k N ; x Alteratively, r beig characterized by 0 r k 1 ad N r mod k, px; A = 1 k µ N/k 1,r x U N/k ϕ k x } + 1 k b kc k µ r i=1 b ic i r+2,k 1 x U N k/k ϕ k x I particular, if N k 1 mod k, the px; A = 1 k µ N/k 1,k 1 x U N/k ϕ k x,,

15 EXPLICIT INVERSE OF A TRIDIAGONAL k TOEPLITZ MATRIX 15 ad the eigevalues of A are the k 1 zeros of 1,k 1 x together with all the solutios of the followig N/k algebraic equatios of degree k ϕ k x =2µ cos jkπ, j =1, 2,, N/k N +1 Proof First, remark that A = D 1 J N D, where D is a diagoal matrix of order N, D := diag d 1,d 2,,d N }, d k+i := i 1 s=1 b s µ 2 1 i k, =0, 1,, N/k, ad J N is the tridiagoal k Toeplitz matrix a 1 1 b 1 c 1 ak 1 b k c k a 1 1 J N := b 1 c 1 C N N ak 1 b k c k a 1 1 b 1 c 1 Now otice that J N is the Jacobi matrix of order N correspodig to the MOPS 1 p x} 0, where p x} 0 is the MOPS geerated by the block of recurrece relatios 31 with a j } ad b j } defied by 52 We have see i the proof of theorem 51 that this sequece p x} 0 is determied by 55 Hece sice A ad J N are similar matrices, ad the characteristic polyomial of J N is 1 N p N x, it follows that px; A = 1 N p N x = 1 N Q N a1,,a k ; x The alterative expressio for px; A stated i the theorem comes ow easily takig ito accout the defiitio of Q N i terms of the Chebyshev polyomials of the secod kid as well as the relatios D k x = 1 k π k ; x, i,j x = 1 j i+1 a1,,a k a i,a i+1,,a j b i c i,b i+1 c i+1,,b j 1 c j 1 ; x The formula i the case N k 1 mod k, ie, r = k 1, is a cosequece of the geeral formula sice k+1,k 1 x 0; ad the algebraic equatios for the determiatio of the eigevalues follows from the trigoometric expressio for the Chebyshev polyomials of the secod kid

16 16 C M DA FONSECA AND J PETRONILHO Remark 62 We have proved the precedig theorem assumig k 3 However, a direct ispectio of the proofs i the previous theorems shows that the secod expressio for px; A ad the formulas after it remais valid for k =1adk =2 provided oe defies ϕ 1 x := 1 2µ x a 1, ϕ 2 x := 1 2µ x a 1x a 2 b 1 c 1 + b 2 c 2 } I fact, this ca easily be see by comparig the relatio 1 k T x =b k c k + µ 2 /b k c k +D k x, which is true for k 3 accordig to 53 with the defiitio 54 of T, which is valid for ay k 1 As a cosequece, we coclude that Theorem 61 recover ad geeralizes results from [4], [7], [13] ad [14], where the characteristic polyomial of A have bee computed is some special situatios I fact, the cases r =0adr = k 1 for arbitrary k have bee treated i [4]; the case k = 2 i [7] ad [13], but we otice that this is a particular situatio of the cases treated i [4]; ad the solutio for the case k = 3 have bee preseted i [15] which oly partially follows from [4] or From Theorem 61 we get det A = 1 N p0; A =Q N a1,,a k ;0, det A = 1 r µ N/k 1,r 0 U N/k ϕ k 0 } + 1 k b kc k µ r i=1 b ic i r+2,k 1 0 U N k/k ϕ k 0 Hece, it follows that A is osigular if ad oly if a1,,a k ;0 0, which, of course, ca be reformulated i terms of the secod expressio for det A I particular, whe N k 1mod N the A 1 exists if ad oly if Q N jkπ 1,k 1 0 0, ϕ k 0 2µ cos, j =1, 2,, N/k N +1 Cosider agai a geeral N ad assume that the coditios 1,r 0 0, ϕ k 0 [ 1, 1] hold the last oe esures that U s ϕ k 0 0 for all s The the quatity ζ N,k A := 1 k+1 b kc k µ r i=1 b ic i r+2,k 10 1,r 0 is effectively computable ad from the above expressio for det A ad the asymptotic result 24 we get that for N large eough A 1 exists provided that ζ N,k A ϕ k 0 + ϕ 2 k 0 1, where the square root is take i the same sese as i 24

17 EXPLICIT INVERSE OF A TRIDIAGONAL k TOEPLITZ MATRIX 17 Refereces [1] J Charris, M E H Ismail, ad S Mosalve, O sieved orthogoal polyomials X: geeral blocks of recurrece relatios, Pac J Math o 2, [2] J Charris ad M E H Ismail, Sieved orthogoal polyomials VII: geeralized polyomial mappigs, Tras Amer Math Soc o 1, [3] T S Chihara, A itroductio to orthogoal polyomials, Gordo ad Breach, New York, 1978 [4] L Elser ad R M Redheffer, Remarks o bad matrices, Numer Math , [5] C M da Foseca ad J Petroilho, Explicit iverses of some tridiagoal matrices, Liear Algebra Appl , 7-21 [6] J Geroimo ad W Va Assche, Orthogoal polyomials o several itervals via a polyomial mappig, Tras Amer Math Soc , [7] M J C Gover, The eigeproblem of a tridiagoal 2 Toeplitz matrix, Liear Algebra Appl 197/ , [8] MJCGoveradSBarett,Iversio of Toeplitz matrices which are ot strogly sigular, IMA J Numer Aal , [9] G Heiig ad K Rost, Algebraic methods for Toeplitz-like matrices ad operators, Birkhäuser OT 13, 1984 [10] M E H Ismail ad Xi Li, O sieved orthogoal polyomials IX: orthogoality o the uit circle, Pac J Math , [11] J W Lewis, Iversio of tridiagoal matrices, Numer Math , [12] R K Mallik, The iverse of a tridiagoal matrix, Liear Algebra Appl , [13] F Marcellá ad J Petroilho, Eigeproblems for tridiagoal 2-Toeplitz matrices ad quadratic polyomial mappigs, Liear Algebra Appl , [14] F Marcellá ad J Petroilho, Orthogoal polyomials ad quadratic trasformatios, Port Math , [15] F Marcellá ad J Petroilho, Orthogoal polyomials ad cubic polyomial mappigs I, Commu Aalytic Theory of Cotiued Fractios, , [16] G Meurat, A review o the iverse of symmetric tridiagoal ad block tridiagoal matrices, SIAM J Matrix Aal Appl, o 3, [17] D Moskovitz, The umerical solutio of Laplace s ad Poisso s equatios, Quart Appl Math, [18] J Petroilho, Variatios o sieved orthogoal polyomials o the uit circle submitted [19] P Schlegel, The explicit iverse of a tridiagoal matrix, Math Comp o 111, 665 [20] R Usmai, Iversio of a tridiagoal Jabobi matrix, Liear Algebra Appl 212/ , [21] T Yamamoto ad Y Ikebe, Iversio of bad matrices, Liear Algebra Appl , Dep de Matemática, FCTUC, Uiv de Coimbra, Coimbra, PORTUGAL address: cmf@matucpt Dep de Matemática, FCTUC, Uiv de Coimbra, Coimbra, PORTUGAL address: josep@matucpt