EXPLICIT EIGENVALUES AND INVERSES OF SEVERAL TOEPLITZ MATRICES

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1 ANZIAM J. 48(2006), EXPLICIT EIGENVALUES AND INVERSES OF SEVERAL TOEPLITZ MATRICES WEN-CHYUAN YUEH 1 ad SUI SUN CHENG 2 (Received 16 February, 2005; revised 3 March, 2006) Abstract Based o the theory of differece equatios, we derive ecessary ad sufficiet coditios for the existece of eigevalues ad iverses of Toeplitz matrices with five differet diagoals. I the course of derivatios, we are also able to derive computatioal formulas for the eigevalues, eigevectors ad iverses of these matrices. A umber of explicit formulas are computed for illustratio ad verificatio Mathematics subect classificatio: primary 15A09, 15A18; secodary 39A10. Keywords ad phrases: Toeplitz matrix, eigevalue, eigevector, iverse, differece equatio, sequece. 1. Itroductio A Toeplitz matrix is a matrix with values costat alog each (top-left to lowerright) diagoal. Several properties of these matrices are ow kow, icludig their eigevalues, eigevectors ad iverses. I particular, i a recet paper [4] by Dow, Toeplitz matrices of the form T = b c Þ a b c a b c 0 0 Þ a b where the corer elemets are the same, are discussed ad their explicit iverses are foud. I may applicatios (such as boudary value problems for differece 1 Departmet of Refrigeratio, Chi-Yi Istitute of Techology, Taichug, Taiwa 411, R. O. Chia; yuehwc@chiyi.cit.edu.tw. 2 Departmet of Mathematics, Tsig Hua Uiversity, Hsichu, Taiwa 30043, R. O. Chia. c Australia Mathematical Society 2006, Serial-fee code /06 ; 73

2 74 We-Chyua Yueh ad Sui Su Cheg [2] equatios), Toeplitz matrices of the form A = T but with the Þ i the bottom lefthad side corer replaced by þ,thatis,[a 1 ]=þ,whereþ = þ, are also ecoutered. Therefore, it is of great iterest to fid out more about these matrices. I this paper, we derive the eigevalues ad their correspodig eigevectors as well as the iverse of A whe ac = 0 ad at least oe of the umbers Þ or þ is ot zero. Whe Þ = þ = 0, A reduces to the well-kow tridiagoal matrix about which much is kow. For geeral iformatio about Toeplitz matrices, the refereces i [4] ca be cosulted. For coveiece, the set of itegers, the set of oegative itegers, the set of real umbers ad the set of complex umbers are deoted by Z, N, R ad C respectively. The umber 1 is deoted by i. We will also set ÞZ =mþ m Z}; Þ C: I particular, ³Z deotes the set :::; 2³; ³; 0;³;2³;:::}. Toeplitz matrices are itimately related to boudary value problems ivolvig differece equatios. This relatioship has bee exploited i [10, 2] for fidig eigevalues or iverses of matrices arisig from differece operators. We will agai base our ivestigatio here o the method of differece equatios. For this reaso, we recall some termiologies used i [1]. Let l N be the set of complex sequeces of the form x =x k } k N edowed with the usual liear structure. A sequece of the form Þ; 0; 0;:::} is deoted by Þ (or by Þ if o cofusio is caused), ad the sequece 0; 1; 0; 0;:::} is deoted by h. Give two sequeces x =x k } ad y =y k } i l N, their covolutio is deoted by x y (or xy if o cofusio is caused) ad is defied by } xy = x k y k : k=0 It is easily verified that h 2 = h h =0; 0; 1; 0; 0;:::} ad h = h } N, = 1; 2;:::,isgiveby h = 1if = ad h = 0 otherwise. We will also set h 0 = 1. I the followig discussios, we eed, amog other thigs, the well-kow properties of the complex fuctios e z,siz ad cos z from the theory of complex aalysis. I particular, let z = x + iy C where x; y R. The (i) si z = 0 if ad oly if y = 0adx = k³ for some k Z, (ii) cos z =±1 if ad oly if y = 0adx = k³ for some k Z, ad (iii) if z = k³ for ay k Z, thesiz = 0, cos z = ±1, si.z=2/ = 0 ad cos.z=2/ = 0. N 2. Necessary coditios for the eigevalues Cosider the eigevalue problem A u = ½u,wherea; b; c ad Þ, þ are umbers i the complex plae C. We will assume that ac = 0. Whe ac = 0, the correspodig

3 [3] Explicit eigevalues ad iverses of several Toeplitz matrices 75 aalysis is quite differet ad is treated elsewhere. To avoid trivial coditios, we will also assume 3 i the followig discussios. Let ½ be a eigevalue (which may be complex) ad.u 1 ;:::;u / a correspodig eigevector of A. We may view the umbers u 1 ; u 2 ;:::;u respectively as the first, secod,..., ad the -th term of a ifiite (complex) sequece u =u i } i=0.sice A u = ½u ca be writte as u 0 = 0; au 0 + bu 1 + cu 2 = ½u 1 Þu ; au 1 + bu 2 + cu 3 = ½u 2 ;... (2.1) au 2 + bu 1 + cu = ½u 1 ; au 1 + bu + cu +1 = ½u þu 1 ; u +1 = 0; we see that the sequece u = u k } k=0 satisfies u 0 = 0; u +1 = 0ad au k 1 + bu k + cu k+1 = ½u k + f k ; k = 1; 2;:::; (2.2) where f 1 = Þu ad f = þu 1, while f k = 0fork = 1;. Note that u ad u 1 caot be 0 simultaeously, for otherwise from (2.1), u 2 = 0 ad iductively u 3 = u 4 = =u = 0, which is cotrary to the defiitio of a eigevector. Let f =f k } k=0 be defied above. The (2.2) ca be expressed as cu k+2 } k=0 + bu k+1} k=0 + au k} k=0 = ½u k+1} k=0 +f k+1} k=0 : By takig the covolutio of the above equatio with h 2 = h h, ad otig that ad hu +1 }= hu 1 ; u 2 ;:::}=0; u 1 ; u 2 ;:::}=u u 0 h 2 u +2 }= h 2 u 2 ; u 3 ;:::}=0; 0; u 2 ; u 3 ;:::}=u u 0 u 1 h; we have c.u u 0 u 1 h/ +.b ½/ h.u u 0 / + a h 2 u = h. f f 0 /. Solvig for u, ad substitutig f 0 = u 0 = 0, we obtai ( a h 2 +.b ½/ h + c ) u =.cu 1 + f / h: Sice c = 0, we ca divide the above equatio by a h 2 +.b ½/ h + c to obtai [1, Theorem 24] u =.cu 1 + f / h a h 2 +.b ½/ h + c : (2.3)

4 76 We-Chyua Yueh ad Sui Su Cheg [4] Let ± =.b ½/ ±! 2a be the two roots of az 2 +.b ½/z + c = 0, where! =.b ½/ 2 4ac, which may or may ot be zero. Basedothevalueof!, there are two cases to be cosidered. Case I. Suppose! = 0 so that + ad are distict. Sice + = c=a = 0, we may write ± = e ±i =² for some i the strip z C 0 Re z < 2³}, where ² = a=c ad cos =.½ b/=2²c : (2.4) Sice si =!=.2i²c/ = 0, we must also have = ³Z. By the method of partial fractios, we ca the write (2.3) i the form u = 1! ( 1 h 1 = 1!. +1/ ).cu 1 + f / h + h }. +1/ +.cu N 1 + f / h = 2i! ² si } cu 1 ; Þu ; 0;:::; þu 1 ; 0;:::}: By evaluatig the covolutio product, we obtai the -th term of u, u = 2i! ( cu1 ² si Þu ² 1 si. 1/ H. /þu 1 ² si. / ) (2.5) for 1, where H.x/ is the uit step fuctio defied by H.x/ = 1ifx 0ad H.x/ = 0ifx < 0. I particular,! 2i u = cu 1 ² si Þu ² 1 si. 1/ (2.6) ad! 2i u +1 = cu 1 ² +1 si. + 1/ Þu ² si þu 1 ² si : (2.7) By (2.6), we have ( )! c².si /u 1 + Þ² 1 si. 1/ u = 0; (2.8) 2i

5 [5] Explicit eigevalues ad iverses of several Toeplitz matrices 77 adby(2.7), ad the coditio u +1 = 0, we have ( c² +1 si. + 1/ þ² si ) u 1 Þ².si /u = 0: (2.9) Sice u 1 ad u caot be both zero, we must ecessarily have (! ) c² si + Þ² 1 si. 1/ 2i c² +1 si. + 1/ þ² si Þ² si = 0; which leads to the ecessary coditio Þc² 2 si ² ( ac si. + 1/ Þþ si. 1/ ) + aþ si = 0: (2.10) Oce we have foud a that satisfies (2.10), we obtai by (2.4) ½ = b + 2²c cos ; = m³; m Z: (2.11) Case II. Suppose! = 0sothat + =. I this case,.½ b/ 2 = 4ac, ad (2.3) ca be writte as.cu 1 + f / h u = c ( 1 ² h 2..½ b/=2c/ h +..½ b/=2c/ 2 2) = h 1 ²c ( 1 ² h ).cu f / = 1 ²c } ² cu N 1; Þu ; 0;:::; þu 1 ; 0;:::} ; where The -th term of u ow becomes ² =.½ b/=2c =± a=c =±²: (2.12) u = 1 ²c ( cu1 ² Þu. 1/ ² 1 H. /þu 1. / ² ) ; 1: I particular, (2.13) ad u = ( cu 1 ² Þu. 1/ ² 1) = ²c (2.14) u +1 = 0 = ( cu / ² +1 Þu ² þu 1 ² ) = ²c: (2.15) This leads to the ecessary coditio Þc ² 2 ² ( ac. + 1/ Þþ. 1/ ) + aþ = 0: (2.16)

6 78 We-Chyua Yueh ad Sui Su Cheg [6] Oce we have foud the ² that satisfies (2.16), the we obtai by (2.12) We remark that sice (2.17) may be writte as ½ = b + 2 ²c: (2.17) ½ = b ± 2²c = b + 2²c cos ; = m³; m Z; we may combie (2.17) ad (2.11) ad assert that a eigevalue ½ of A is ecessarily of the form ½ = b + 2²c cos. Accordig to the above discussios, whe ½ is a eigevalue of A, it is the ecessary that either (2.10) or(2.16) holds. THEOREM 2.1. Let ½ be a eigevalue of the matrix A ad u =.u 1 ;:::;u / its correspodig eigevector. If (2.10) is satisfied for some the (2.5) ad (2.11) hold. z C 0 Re z < 2³}\³Z; THEOREM 2.2. Let ½ be a eigevalue of the matrix A ad u =.u 1 ;:::;u / its correspodig eigevector. If (2.16) is satisfied for ² = a=c or ² = a=c, the (2.13) ad (2.17) hold. Recall that the first ad last compoets u 1 ad u caot be zero simultaeously. There are some other iterestig properties for the eigevector u if! = 0. COROLLARY 2.3. Let ½ be a eigevalue of the matrix A ad u =.u 1 ;:::;u / its correspodig eigevector such that! =.b ½/ 2 4ac = 0. Let be the umber foud i Theorem 2.1. (i) If u 1 = 0, the Þ = 0. (ii) If u = 0 or u 1 = 0 the si = 0. (iii) If si = 0, the either u = 0 or Þ =±a², ad either u 1 = 0 or þ =±c². (iv) If þ = ±c²,theu = 0. (v) If Þ = ±a²,theu 1 = 0. (vi) If þ = ±c² ad u 1 = 0, the si = 0. (vii) If Þ = ±a² ad u = 0, the si = 0. (viii) If þ = ±c² ad Þ = ±a²,theu 1 = 0; u = 0 ad si = 0. PROOF. If u 1 = 0, the by (2.8)!=.2i/ + Þ² 1 si. 1/ = 0. Sice! = 0, we must have Þ = 0.

7 [7] Explicit eigevalues ad iverses of several Toeplitz matrices 79 If u = 0, the by (2.8) c² u 1 si = 0. Sice c² u 1 = 0, we must have si = 0. Similarly, if u 1 = 0, the by (2.9) Þ² u si = 0. Sice Þ² u = 0, we must have si = 0. If si = 0, the by (2.8), either u = 0or²csi + Þ² 1 si. 1/ = 0. Sice si. 1/ = si cos cos si = si whe si = 0; the latter implies ²c =±Þ² 1 or Þ =±a². Similarly, if si = 0, the by (2.9), either u 1 = 0orc² +1 si. + 1/ þ² si = 0. The latter implies þ =±c². Suppose þ = ±c². If u = 0, the si = 0. Sice u 1 = 0, (iii) implies þ =±c², which is a cotradictio. Suppose Þ = ±a². If u 1 = 0, the si = 0. Sice u = 0, (iii) implies Þ =±a², which is a cotradictio. Suppose þ = ±c² ad u 1 = 0. If si = 0, the by (iii), either u 1 = 0or þ =±c². This is a cotradictio. Suppose Þ = ±a² ad u = 0. If si = 0, the by (iii), either u = 0or Þ =±a². This is a cotradictio. The last assertio (viii) follows from (iv), (v) ad (vi), (vii). 3. Additioal coditios for eigevectors Giveaeigevalue½ of the matrix A ad its correspodig eigevector u =.u 1 ;:::;u /, suppose! = 0 ad let be the umber foud i Theorem 2.1. For 1,wehaveby(2.5) 1 ( u = cu1 ² si Þu ² 1 si. 1/ ) ; = 1;:::;: (3.1) ²c si I the case whe si = 0, we may fid u which are simpler i form. Ideed, suppose si = 0, the from (2.8), ad from (2.9), u 1 =.²c si + Þ² 1 si. 1// c² si Þu =.c²+1 si. + 1/ þ² si / ² si u (3.2) u 1 : (3.3) Substitutig (3.2) ito(3.1), we have ( 1 ² ) si u = ²c si ² si.²c si + Þ² 1 si. 1//u Þu ² 1 si. 1/ u ( = ²c² si si + Þ² 1 si. / si ) ²c si si

8 80 We-Chyua Yueh ad Sui Su Cheg [8] for = 1; 2;:::;. By lettig u = ² si, we obtai u = ² si + Þ a ² + si. /; = 1;:::;; (3.4) which defies a eigevector correspodig to ½ (if is foud). Similarly, substitutig (3.3)ito(3.1), we may obtai u = ² si. + 1 / + þ c ² si. 1/; = 1;:::;: (3.5) Suppose! = 0. The we have by (2.13) u = 1 ²c ( cu1 ² Þu. 1/ ² 1) ; = 1;:::;: I view of (2.14) ad (2.15), a similar argumet leads to ad u = ² + ² Þ +. /; = 1;:::; (3.6) a u = ². + 1 / + ² þ. 1/; = 1;:::;: (3.7) c 4. Eigevalues ad eigevectors of special Toeplitz matrices Now we ca apply the results of the previous sectios to fid the eigevalues ad the correspodig eigevectors of several Toeplitz matrices of the form A.For motivatio, cosider the case where Þ = þ = 0. The (2.16) is reduced to ² ac. + 1/ = 0 which is ot possible so that, i view of Theorem 2.2,! = 0ad½ caot be of the form b ± 2²c. I view of Theorem 2.1, (2.10) must hold for some = ³Z, or, si. + 1/ = 0forsome = ³Z. Cosequetly, = k³=. + 1/ for some k Z ad = ³Z. A eigevalue ½ k is the ecessarily of the form k³ ½ k = b + 2²c cos + 1 ; k = 1;:::;: A correspodig eigevector, from (3.4), is give by u.k/ = ² si k³ ; = 1;:::;; (4.1) + 1 which has also bee obtaied i [10] ad elsewhere. Fially, by reversig the argumets that lead to Theorem 2.1, we see that each ½ k is a eigevalue of A ad the correspodig vector u.k/ =.u.k/ 1 ;:::;u.k/ / defied by (4.1) is a correspodig eigevector. By similar ideas, we may ow derive the eigevalues ad the correspodig eigevectors for matrices of the form A.

9 [9] Explicit eigevalues ad iverses of several Toeplitz matrices The case where a = c =±α ad β = 0, or α = 0ada = c =±β We will use [x] to deote the itegral part of x R. Note that [.k 1/=2] +[.k + 2/=2] = k for ay positive iteger k. THEOREM 4.1. Suppose Þ = a = c ad þ = 0, or,þ = 0 ad þ = c = a. The the eigevalues of A are give by ½ k = b + 2a cos 2k³ ; k = 1; 2;:::;[. 1/=2] (4.2) ad.2m 1/³ ½ m+[. 1/=2] = b + 2a cos ; m = 1; 2;:::;[. + 2/=2]: (4.3) + 2 The eigevectors correspodig to (4.2) ad (4.3) are give by ad u.k/ u.m+[. 1/=2]/ = si = si respectively for Þ = a = c ad þ = 0, ad u.k/ = si 2k³ ad respectively for Þ = 0 ad þ = c = a. 2k. + 1 /³ ; = 1; 2;:::; (4.4).2m 1/. + 1 /³ ; = 1; 2;:::; (4.5) + 2 u.m+[. 1/=2]/ = si.2m 1/ ³ ; = 1; 2;:::; + 2 PROOF. Suppose Þ = a = c ad þ = 0, or Þ = 0adþ = c = a. The (2.16) is reduced to.±1/. + 1/ = 1. This relatio caot be valid, ad hece, i view of Theorem 2.2,! = 0 does ot hold. I view of Theorem 2.1, (2.10) holds for some = ³Z,or si ( ) cos = 0; = ³Z: 2 Hece (a) si.=2/ = 0 or (b) cos.. + 2/=2/ = 0forsome = ³Z. I case (a), we have so that a eigevalue must be of the form Similarly, i case (b), we have = 2k³=; = ³Z; k Z; (4.6) ½ k = b + 2a cos.2k³=/; k = 1;:::;[. 1/=2]: =.2m 1/³ ; = ³Z; m Z; (4.7) + 2

10 82 We-Chyua Yueh ad Sui Su Cheg [10] so that a eigevalue must be of the form.2m 1/³ ½ m+[. 1/=2] = b + 2a cos ; m = 1;:::;[. + 2/=2]: + 2 The correspodig eigevectors may be obtaied as follows. For Þ = a = c ad þ = 0, sice þ = ±c², by Corollary 2.3 (iv) u = 0, while u 1 may or may ot be 0. If u 1 = 0, the si = 0. By (3.1) u = u si. 1/: si Sice si. +1 / = si cos. 1/ cos si. 1/ = si. 1/ whe si = 0, we may write u =.±u =si /si. + 1 /. Lettig ±u = si, we have u = si. + 1 /; = 1; 2;:::;: (4.8) If u 1 = 0, the by Corollary 2.3 (vi), si = 0. Hece we may apply (3.5), which leads to the same result (4.8) sice ² = 1 ad þ = 0. By substitutig give by (4.6) or (4.7)ito(4.8), we obtai the desired results (4.4) ad (4.5). For Þ = 0adþ = c = a, a similar argumet leads to u = si, = 1;:::;. By substitutig giveby(4.6)or(4.7), we obtai the desired results. Oce we have foud the eigevalues ad their correspodig eigevectors, we may reverse the argumets leadig to Theorem 2.1 ad verify that they are ideed the true eigevalues ad associated eigevectors of A. The proof is complete. We may follow the same argumets to show the followig. Suppose Þ = a = c ad þ = 0, or Þ = 0ad þ = c = a. The the eigevalues of A are give by ad ½ k = b + 2a cos 2k³ ; k = 1; 2;:::;[. + 1/=2] (4.9) + 2.2m 1/³ ½ m+[.+1/=2] = b + 2a cos ; m = 1; 2;:::;[=2]: (4.10) The eigevectors correspodig to (4.9) ad (4.10) are give by ad u.k/ u.m+[.+1/=2]/ = si = si 2k. + 1 /³ + 2.2m 1/. + 1 /³ ; = 1; 2;:::; ; = 1; 2;:::;;

11 [11] Explicit eigevalues ad iverses of several Toeplitz matrices 83 respectively for Þ = a = c ad þ = 0, ad ad u.k/ u.m+[.+1/=2]/ = si = si 2k³ + 2 ; respectively for Þ = 0ad þ = c = a. = 1; 2;:::;.2m 1/ ³ ; = 1; 2;:::; 4.2. The case where α = β = a = c,or α = β = a = c THEOREM 4.2. Suppose Þ = þ = a = c, or Þ = þ = a = c. The the eigevalues of A are give by b + 2a cos.k³=/; k = 1; 2;:::; 1; ½ k = b; k = : (4.11) The eigevectors correspodig to (4.11) are give by ad si. k³=/; k odd, u.k/ = si.. 1/k³=/; k eve, u./ = si. ³=2/ +.Þ=a/ si.. /³=2/; odd, si. ³=2/; = 6; 10; 14;::: si.. 1/³=2/; = 4; 8; 12;::: (4.12) (4.13) respectively for Þ = þ = a = c, = 1; 2;:::;. For Þ = þ = a = c, oly the odd-eve relatio for k i (4.12) should be iterchaged. PROOF. Suppose Þ = þ = a = c, or Þ = þ = a = c, the ² = 1, ad (2.16) is reduced to.±1/ 2 = 0: This relatio caot be valid so that! = 0 does ot hold. By Theorem 2.1, (2.10) holds for some = ³Z, orsi cos = 0, = ³Z. Ithe case where si = 0forsome = ³Z, wehave =.k³=/ = ³Z, k Z, ad the eigevalue must be of the form ½ k = b + 2a cos.k³=/, k = 1; 2;:::; 1. I the case where cos = 0, we have ½ = b. The correspodig eigevectors may be foud as follows. For k = 1;:::; 1, suppose Þ = þ = a = c. Sicesi = si k³ = 0 ad cos = cos k³ = 1 if k is odd ad +1 ifk is eve, we have c si + Þ si. 1/ = a si.1 cos / = 2a si = 0; k odd, ad c si. + 1/ þ si = c.cos + 1/ si = 2a si = 0; k eve.

12 84 We-Chyua Yueh ad Sui Su Cheg [12] Hece if k is odd, by (2.8) u = 0 ad i view of (3.1) a eigevector must be of the form u = si : (4.14) If k iseve,theby(2.9) u 1 = 0 ad a eigevector must be of the form u = si. 1/: (4.15) By substitutig = k³=,wehave(4.12). Suppose Þ = þ = a = c, the(4.14) is for eve k ad (4.15) is for odd k. For k =, wehave = ³=2 ad si = si.³=2/ =±1 = 0if is odd. We may apply (3.4) to obtai u./ = si ³ 2 + Þ a. /³ si ; = 1; 2;:::;: 2 If iseve,thesi = 0 ad cos = 1if = 6; 10;::: ad +1if = 4; 8;:::. Hece by a similar argumet as for k = 1; 2;:::; 1, we have for = 1;:::; u./ = The proof is complete. si. ³=2/; for = 6; 10; 14;:::; si.. 1/³=2/; for = 4; 8; 12;:::: 4.3. The case where α =±a ad β =±c I the case where Þ = a ad þ = c, A is the well-kow circulat matrix [3]. There are may results [3, 7, 8, 9, 5, 6] cocerig the eigevalues ad iverses of such matrices. However, most of them are algorithmic i ature. Here we will derive explicit formulas for the case where Þ = a ad þ = c based o our theorems, while those for Þ = a ad þ = c will be listed oly sice they are already kow. THEOREM 4.3. Suppose Þ = a ad þ = c i the matrix A. The the eigevalues ad the correspodig eigevectors of A are give by ad.2k 1/³ ½ k = b +.a + c/ cos.2k 1/³ + i.a c/ si ; k = 1; 2;:::; (4.16) u.k/ = e i.2k 1/³= ; = 1; 2;:::; (4.17)

13 [13] Explicit eigevalues ad iverses of several Toeplitz matrices 85 respectively. If a = c, a alterative formula for the eigevectors is also give by u.k/.2k 1/ ³ = c 1 cos + c 2 si.2k 1/ ³ ; = 1; 2;:::;; (4.18) where c 1 ad c 2 are two idepedet costats ot both equal to 0: I particular, if we take c 1 = 1 ad c 2 = i, we have (4.17) as its special case. PROOF. Suppose the coditios i Theorem 2.1 hold. The we must have ½ = b + 2²c cos,where is some umber that satisfies ² ( si. + 1/ si. 1/ ) +.² 2 + 1/ si = 0; = ³Z; or, sice si = 0, ² 2 + 2² cos + 1 = 0, = ³Z. This yields ² = cos ± i si = e i : (4.19) Let ² = e i,thesicee ±i.2k 1/³ = 1, we may write ² = e i e i.2k 1/³,sothat ad e i = ²e i.2k 1/³= ; e i = ² 1 e i.2k 1/³= (4.20) cos = 1 ( ²e i.2k 1/³= + ² 1 e i.2k 1/³=) 2 = 1 [( ² + 1 ).2k 1/³ cos + i 2 ² ( ² 1 ) si ² ].2k 1/³ : Note that if a = c, the ² = 1 ad cos = ±1 for ay k Z. By otig that ² 2 c = a ad that si x ad cos x are periodic fuctios, we have fially for k = 1;:::; ½ k = b + 2²c cos = b +.a + c/ cos.2k 1/³ + i.a c/ si.2k 1/³ which is (4.16). If a = c, the ² = 1 ad we have cos = cos..2k 1/³=/ = ±1 so that k =. + 1/=2i(4.16). But the we have Theorem 2.2. Suppose! = 0 ad the coditios i Theorem 2.2 hold. The (2.16) is valid for Þ = a ad þ = c. Thus ² ² + 1 = 0; (4.21) which holds if a = c, ² = 1ad is odd. Furthermore, uder these coditios, the eigevalue must be of the form ½ = b 2c, which ca also be writte as ½.+1/=2 i (4.16). Hece (4.16) holds regardless of a = c or a = c. ;

14 86 We-Chyua Yueh ad Sui Su Cheg [14] I case the egative sig i e i holds, the ² = e i = e i e i.2k 1/³,adit is easily see that we may get the same result (4.16). To fid the correspodig eigevectors, we first cosider the case a = c. The ² = 1 so that þ = ±c² ad Þ = ±a². By Corollary 2.3 (viii), u 1 ; u 2 = 0ad si = 0, so we may apply (3.4). Sice by (4.20) e ±i = ² ± e ±i.2k 1/³=,wehave e ±i. / = ² ±. / e ±i. /.2k 1/³= = ² ±. / e i.2k 1/³= : By substitutig this ad Þ = a ito (3.4), u.k/ = 1 ( ( ² ² e i.2k 1/³= ² e i.2k 1/³=) 2i +² ( + ² e i.2k 1/³= ² + e i.2k 1/³=)) = 1 2i.²2 1/e i.2k 1/³= : By droppig the costat factor.² 2 1/=2i, weobtai(4.17). Next suppose a = c. By (4.19), ² = 1 implies cos = 1ad si = 0. The former implies =.2k 1/³= = ³Z so that k =. + 1/=2, the latter implies either oe of the u 1 or u may be zero. If u = 0, the by (3.1) a eigevector must be of the form u.k/ If u 1 = 0, the a eigevector must be of the form u.k/ liear combiatio u.k/ = k 1 si + k 2 si. 1/ = c 1 cos + c 2 si = si. = si. 1/. Hece the is a eigevector of A correspodig to ½ k. After substitutig =.2k 1/³=, we have (4.18)fora = c ad k =. + 1/=2. For a = c ad k =. + 1/=2, the = ³, which implies! = 0, ad we already have ² = ² = 1 from (4.21), hece we may apply either (3.6)or(3.7) to obtai u =. 1/ ; = 1;:::;; which is of the form u.+1/=2 i (4.18). Hece (4.18) is valid for a = c ad k = 1;:::;. The proof is complete. Now we may follow the same argumets to show the followig: Suppose Þ = a ad þ = c i the matrix A, the the eigevalues ad the correspodig eigevectors of A are give by ad ½ k = b +.a + c/ cos 2k³ u.k/ + i.a c/ si 2k³ ; k = 1; 2;:::; 2 k³ i = e ; = 1; 2;:::; (4.22)

15 [15] Explicit eigevalues ad iverses of several Toeplitz matrices 87 respectively. If a = c, a alterative formula for the eigevectors is also give by [2] u.k/ = c 1 cos 2 k³ + c 2 si 2 k³ ; = 1; 2;:::;; where c 1 ad c 2 are two idepedet costats ot both equal to 0: I particular, if we take c 1 = 1adc 2 = i,wehave(4.22) as its special case. 5. Necessary coditios for the iverse The method used i the previous sectios may also be used to fid the iverse of the matrix A uder the coditio ac = 0. Let the (uique) iverse of A, if it exists, be deoted by g.1/ G = ( 1 g.2/ 1 g./ 1 g.1/ g.2/ g./) = g.1/ 2 g.2/ 2 g./ 2 : (5.1) g.1/ g.2/ g./ The A G = I. We may view the umbers 1 ; 2 ;:::; respectively as the first, secod,..., ad the -th term of a ifiite (complex) sequece = } N. Sice A G = I ca be expaded as with 0 = +1 = 0, we have a 0 + b 1 + c 2 = h k 1 Þg.k/ ; a 1 + b 2 + c 3 = h k ; 2 a 2 + b 3 + c 4 = h k ; 3... a 1 + b + c +1 = h k þg.k/ 1 ; a 1 + b + c +1 = h k + f.k/ ; = 1; 2;::: where Þ = 1; f.k/ = þ 1 = ; 0 otherwise: Sice c = 0, we may obtai =.cg.k/ 1 + h k + f.k/ / h : (5.2) a h 2 + b h + c

16 88 We-Chyua Yueh ad Sui Su Cheg [16] Let ± = ( b ± ¾ ) =2a be the two roots of az 2 + bz + c = 0, where ¾ = b 2 4ac. As i Sectio 2, there are two cases to be cosidered. Case I. Suppose ¾ = 0sothat + ad are two differet umbers. Sice + = c=a = 0, we may write ± = e ±i =² for some i the strip z C 0 Re z < 2³}, where ² = a=c ad cos = b=2²c: (5.3) We also have si = ¾=.2i²c/ = 0. By the method of partial fractios, we may write i the form = 1 ( 1 ¾ h 1 ) (cg.k/ 1 + h k + f.k/) h; + h which gives the -th term of : = 2i ¾ cg.k/ 1 ² si Þ ² 1 si. 1/ + H. k/² k si. k/ H. /þ 1 ² si. / } (5.4) for 1. I particular, ¾ = c ad 2i g.k/ 1 ² si Þ ² 1 si. 1/ + ² k si. k/ 0 = c 1 ² +1 si. + 1/ Þ ² si þ 1 ² si + ² +1 k si. + 1 k/: If the iverse exists, the 1 ad g.k/ form a uique solutio pair ad hece 1 = c² si ( ¾=2i + Þ² 1 si. 1/ ) c² +1 si. + 1/ þ² si Þ² si = 0; or 1 = ( ².ac si. +1/ Þþ si. 1//.Þc² 2 +aþ/si ) si = 0: (5.5) Furthermore, if 1 = 0, the we have 1 = 1 1 =1; = 1 =1; (5.6) where 1 1 = Þ² 2 k si.k 1/ si a² k si. + 1 k/ si ad (5.7) 1 = c² 2+1 k si k si þ² +1 k si. k/ si : (5.8)

17 [17] Explicit eigevalues ad iverses of several Toeplitz matrices 89 Case II. Suppose ¾ = 0 so that ± are two equal roots. I this case b 2 = 4ac. Furthermore, from (5.2), we have.c 1 + h k + f.k/ / h = c.1 2. b=2c/ h +. b h=2c/ 2 / = 1 ²c ² h.1 ² h/ 2.cg.k/ 1 + h k + f.k/ / = 1 ²c ² }.c 1 + h k + f.k/ /; where ² = b=2c. The-th term of is ow = 1 cg.k/ 1 ² Þ 1. 1/² ²c I particular, 0 = 1 ( cg.k/ ²c + H. k/. k/² k H. /þ 1. /² } : (5.9) = ( c 1 ² Þ. 1/² 1 +. k/² k) =²c ad /² +1 Þ ² k/² +1 k þ 1 ² ) : If the iverse exists, the 1 ad form a uique solutio pair ad hece we must have ad where 1 = Þc² 2 + ² ( ac. + 1/ Þþ. 1/ ) aþ = 0; (5.10) 1 = 1 1 =1; = 1 =1; (5.11) 1 1 = ² 2 k Þ.k 1/ ² k a. + 1 k/ ad (5.12) 1 = ² 2+1 k ck ² +1 k þ. k/: (5.13) THEOREM 5.1. Let the iverse of the matrix A be deoted by G = ( g.1/ g./) : If b 2 4ac = 0, the the ecessary ad sufficiet coditio for the iverse to exist is that (5.5) holds for some z C 0 Re z < 2³} that satisfies (5.3). Furthermore, if the iverse exists, the, 2 1, are give by (5.4), while 1 ad g.k/ are give by (5.6). If b 2 4ac = 0, the the ecessary ad sufficiet coditio for the iverse to exist is that (5.10) holds. Furthermore, if the iverse exists, the, 2 1, are give by (5.9), while 1 ad g.k/ are give by (5.11).

18 90 We-Chyua Yueh ad Sui Su Cheg [18] We remark that sufficiet coditios for the existece of the iverse of A are added i the above result. This is valid sice the above argumets ca be reversed. We remark also that sice cos z is 2³-periodic, the restrictio z C 0 Re z < 2³} ca be relaxed to C. Furthermore, sice if b 2 = 4ac, thecos = ±1ad= ³Z automatically (cf. Theorem 2.1). 6. Iverses of some special Toeplitz matrices We may ow apply Theorem 5.1 for fidig the iverses for several special Toeplitz matrices. For motivatio, cosider the case where Þ = þ = 0iA.Let be the k-th colum i the iverse G of A. If b 2 = 4ac, the by substitutig Þ = þ = 0 ito (5.10) to(5.12), we have 1 = ² ac. + 1/ = 0ad1 1 = ² k a. + 1 k/. Substitutig these ito (5.9), we obtai = 1 ( cg.k/ 1 ² + H. k/² k. k/ ) ²c = ² k ²c. + 1/. + 1 k/; < k; k. + 1 /; k: After fidig, we may directly reverse the argumets leadig to Theorem 5.1 ad coclude that G = ( g.1/ g.2/ g./). Now let us suppose that b 2 = 4ac. Suppose also that the iverse G of A exists ad is of the form (5.1). The substitutig Þ = þ = 0ito(5.5) (5.7), i view of Theorem 5.1, we ecessarily have cos = b=2²c, C, 1 = ² ac si. + 1/ si = 0 ad 1 1 = a² k si. + 1 k/ si : Substitutig these ito (5.4), we have = = ( 1 ² k si si. + 1 k/ ²c si si. + 1/ ² k ²c si si. + 1/ ) + H. k/² k si. k/ si si. + 1 k/; < k; si k si. + 1 /; k: Oce we have foud,theifsi. + 1/ = 0, we may reverse the argumets leadig to Theorem 5.1 ad coclude that ( g.1/ g.2/ g./) is our desired iverse. O the other had, if si. + 1/ = 0, the 1 = 0 ad by Theorem 5.1, the iverse of A does ot exist.

19 [19] Explicit eigevalues ad iverses of several Toeplitz matrices The case where α =±a ad β = 0 THEOREM 6.1. Suppose Þ = a ad þ = 0 i the matrix A. (i) Suppose b 2 = 4ac. The the iverse G of A give by (5.1) exists if, ad oly if, cos = b=2²cforsome C ad si. + 1/ ² si = 0. Furthermore, if it exists, the ² k = ²c si.si. + 1/ ² si / si si. + 1 k/ + ² si.k / si ; < k; si k si. + 1 /; k: (6.1) (ii) Suppose b 2 = 4ac. The the iverse G of A give by (5.1) exists if, ad oly if, + 1 ² = 0 ad, if it exists, = ² k ²c. + 1 ² /. + 1 k/ + ².k /; < k; k. + 1 /; k: (6.2) PROOF. Suppose the iverse of A exists ad is of the form (5.1). If b 2 = 4ac, the substitutig Þ = a ad þ = 0 ito(5.5) (5.8), we ecessarily have cos = b=2²c, C, 1 = ac².si. + 1/ ² si / si = 0; 1 1 = a² k.² si.k 1/ + si. + 1 k// si ad 1 = c² 2+1 k si k si : Substitutig these ito (5.4), we obtai = ² ( k ² si. k/ si si. + 1 k/ si ²c si si. + 1/ ² si ) + H. k/ si. k/ ; which is equivalet to (6.1). Oce we have foud,theifsi. + 1/ ² si = 0, we may verify that G is the iverse of A. O the other had, if si. + 1/ ² si = 0, the the iverse does ot exist. If b 2 = 4ac, the by substitutig Þ = a;þ = 0ito(5.10)to(5.13), we ecessarily have 1 = ac². + 1 ² / = 0, 1 1 = a² k.².k 1/ k// ad 1 = ² 2+1 k ck. Substitutig these ito (5.9), we obtai = ² } k ². k/. + 1 k/ + H. k/. k/ ; ²c + 1 ²

20 92 We-Chyua Yueh ad Sui Su Cheg [20] which is equivalet to (6.2). Oce we have foud,theif + 1 ² = 0, we may verify that G is the iverse of A. O the other had, if + 1 ² = 0, the the iverse does ot exist. The proof is complete. Suppose Þ = a ad þ = 0 i the matrix A. We may follow the same argumets to show the followig. (i) Suppose b 2 = 4ac, the the iverse G of A giveby(5.1) exists if, ad oly if, cos = b=2²c for some C ad si. +1/ +² si = 0. Furthermore, if it exists, the ² k = ²c si.si. + 1/ + ² si / si si. + 1 k/ ² si.k / si ; < k; (6.3) si k si. + 1 /; k: (ii) Suppose b 2 = 4ac, the the iverse G of A give by (5.1) exists if, ad oly if, ² = 0 ad, if it exists, ² k. + 1 k/ = ²c ² / ².k /; < k; k. + 1 /; k: As we have see, the derivatio of the explicit formulas from Theorem 5.1 are straightforward. Theoretically, we ca obtai formulas for arbitrary Þ ad þ, though i most cases those formulas may be complicated i form. But at least we ca obtai elegat formulas for some special combiatios of Þ ad þ. Some of the results are preseted below without proof for compariso ad quick referece. The proof is simple ad may be obtaied i a way similar to that of Theorem The case where α = 0adβ =±c THEOREM 6.2. Suppose Þ = 0 ad þ = c i the matrix A. (i) Suppose b 2 = 4ac. The the iverse G of A give by (5.1) exists if, ad oly if, cos = b=2²cforsome C ad ² si. + 1/ si = 0. Furthermore, if it exists, the ² k = ²c si.si. + 1/ ² si / si si. + 1 k/; < k; si k si. + 1 / + ² si. k/ si ; k: (ii) Suppose b 2 = 4ac. The the iverse G of A give by (5.1) exists if, ad oly if, ². + 1/ 1 = 0 ad, if it exists,. + 1 k/; < k; = ² k ²c. + 1 ² / k. + 1 / + ². k/; k:

21 [21] Explicit eigevalues ad iverses of several Toeplitz matrices 93 Suppose Þ = 0 ad þ = cithe matrix A. (i) Suppose b 2 = 4ac. The the iverse G of A give by (5.1) exists if, ad oly if, cos = b=2²cforsome C ad ² si. + 1/ + si = 0. Furthermore, if it exists, the ² k = ²c si.si. + 1/ + ² si / si si. + 1 k/; < k; si k si. + 1 / ² si. k/ si ; k: (ii) Suppose b 2 = 4ac. The the iverse G of A give by (5.1) exists if, ad oly if, ². + 1/ + 1 = 0 ad, if it exists, ² k. + 1 k/; < k; = ²c ² / k. + 1 / ². k/; k: 6.3. The case where α =±a, β =±c THEOREM 6.3. Suppose Þ = a ad þ = c i the matrix A. (i) Suppose b 2 = 4ac. The the iverse G of A give by (5.1) exists if, ad oly if, cos = b=2²c forsome C ad ² 2 2² cos + 1 = 0. Furthermore, if it exists, the = I particular, whe a = c, ² k ²c si.² 2cos + ² / ² si.k / + si. + k/; < k; ² si. k/ + si. + k /; k: = si. k / + si. k / : 2a si.1 cos / (ii) If b 2 = 4ac ad ² = 1, the the iverse G give by (5.1) exists ad = ² k ²c.² 2 + ² / ².k / +. + k/; < k; ². k/ +. + k /; k: I particular, whe b = 2a = 2c ad is odd, the =. 1/ k. 2 k /=4a: If b = 2a = 2c ad is eve, or b = 2a = 2c, the ² = 1 ad the matrix A is sigular.

22 94 We-Chyua Yueh ad Sui Su Cheg [22] Suppose Þ = a ad þ = cithe matrix A. (i) Suppose b 2 = 4ac. The the iverse G of A give by (5.1) exists if, ad oly if, cos = b=2²c forsome C ad ² 2 + 2² cos + 1 = 0. Furthermore, if it exists, the I particular, whe a = c, (ii) ² k = ²c si.² + 2cos + ² / ² si.k / si. + k/; < k; ² si. k/ si. + k /; k: = si. k / si. k / : 2a si.1 + cos / If b 2 = 4ac ad ² = 1, the the iverse G of A exists ad ² k = ²c.² ² / ².k /. + k/; < k; ². k/. + k /; k: I particular, if b = 2a = 2c, the If b = 2a = 2c ad is eve, the =.2 k /=4a: (6.4) =. 1/ k. 2 k /=4a: (6.5) If b = 2a = 2c ad is odd, the ² = 1 ad the matrix A is sigular. 7. Examples We give two applicatios of our explicit formulas. EXAMPLE 1. I the sychroisatio problems of artificial eural etworks [11], we ecouter the tridiagoal Toeplitz matrix with a = c = 1, b = 2with the corers Þ = þ = 1. Theorem 4.3 gives the eigevalues ad eigevectors for such matrices. As a umerical example, cosider the followig matrix: A 4 = : (7.1)

23 [23] Explicit eigevalues ad iverses of several Toeplitz matrices 95 By Theorem 4.3,wehave ½ k = 2 + 2cos.2k 1/³ ; k = 1; 2; 3; 4; 4 which gives ½ 1 = ½ 4 = 2 + 2ad½ 2 = ½ 3 = 2 2. Sice a = c, the eigevectors may be obtaied by either (4.17)or(4.18). By (4.17), we have u.1/ = ( ) i 2 ; i; 2 i 2 ; 1 : By (4.18), if we take c 1 = 1adc 2 = 0, the u.1/ =. 2=2; 0; 2=2; 1/ ;ifwe take c 1 = 0adc 2 = 1, the u.1/ =. 2=2; 1; 2=2; 0/. It ca be easily checked that they are the correct eigevectors correspodig to ½ 1. We remark that our theorem is also applicable whe b = 2. A ispectio of (4.16) to (4.18) reveals that oly ½ k depeds o b. Thus the eigevalues of the matrix B 4 = (7.2) are give by ½ 1 = ½ 4 = 2 + 2ad½ 2 = ½ 3 = 2 2, with the correspodig eigevectors uchaged. The iverses of (7.1) ad (7.2) may be obtaied by (6.4) ad (6.5) i Theorem 6.3 as A 1 4 = 1 2 respectively ad B 1 4 = EXAMPLE 2. Cosider the followig three-poit boudary value problem [12] of the form 1 2 u k 1 + u k = f.u k /; k = 1; 2;:::;; (7.3) u 0 = Þu l ; u +1 = 0: I matrix form, this may be writte as A u = F.u/;

24 96 We-Chyua Yueh ad Sui Su Cheg [24] where u =.u 1 ;:::;u /, F.u/ =. f.u 1 /;:::; f.u // ad Þ A = : : : : : : : : A ecessary coditio for the system (7.3) to be solvable is that A is ivertible. I the case where l = ad Þ =±1, Theorem 6.1 offers some help. As a umerical example for l = = 6adÞ = 1, sice ² = 1, = cos = ³ ( ad si ³ ) ( si 7³ si ³ ) = 3 = 0; 3 2 we see that the iverse exists ad is give by (6.3)as = 2 si ³.7 k/³.k /³ si si si ³ si k³.7 /³ si 3 3 which yields : < k; k; = : Refereces [1] S. S. Cheg, Partial Differece Equatios (Taylor ad Fracis, Lodo, 2003). [2] S. S. Cheg ad L. Y. Hsieh, Iverses of matrices arisig from differece operators, Util. Math. 38 (1990) [3] P.J.Davis,Circulat Matrices (Wiley ad Sos, New York, 1979). [4] M. Dow, Explicit iverses of Toeplitz ad associated matrices, ANZIAM J. 44 (2003) [5] I. C. Gohberg ad A. A. Semecul, O the iversio of fiite Toeplitz matrices ad their cotiuous aalogs, Mat. Issled. 7 (1972) , (i Russia).

25 [25] Explicit eigevalues ad iverses of several Toeplitz matrices 97 [6] R. T. Gregory ad D. Karey, A collectio of matrices for testig computatioal algorithms (Wiley- Itersciece, New York, 1969). [7] S. B. Haley, Solutios of bad matrix equatios by proectio-recurrece, Liear Algebra Appl. 32 (1980) [8] A. S. Householder, The Theory of Matrices i Numerical Aalysis (Dover Publicatios, New York, 1975). [9] W. F. Trech, A algorithm for the iversio of fiite Toeplitz matrices, SIAM J. Appl. Math. 12 (1964) [10] W. C. Yueh, Eigevalues of several tridiagoal matrices, Appl. Math. E-Notes 5 (2005) [11] W. C. Yueh ad S. S. Cheg, Sychroizatio i a artificial eural etwork, Chaos Solitos Fractals preprit. [12] G. Zhag ad R. Media, Three-poit boudary value problems for differece equatios,comput. Math. Appl. 48 (2004)